diff options
-rwxr-xr-x | skripti/365_prenesi_omejene_vsebine.sh | 50 | ||||
-rw-r--r-- | skripti/completion.bash | 6 | ||||
-rwxr-xr-x | skripti/earhorn_m3u.sh | 43 | ||||
-rwxr-xr-x | skripti/init_script_interpreter.sh | 115 | ||||
-rw-r--r-- | skripti/rš_glasbena_oprema.py | 31 | ||||
-rwxr-xr-x | skripti/service.sh | 28 | ||||
-rwxr-xr-x | skripti/stream_archive.sh | 37 | ||||
-rwxr-xr-x | skripti/videonadzor.sh | 100 | ||||
-rwxr-xr-x | skripti/zone_axfr.py | 86 | ||||
-rwxr-xr-x | skripti/zone_update.py | 174 | ||||
-rw-r--r-- | skripti/ž | 36 | ||||
-rw-r--r-- | šola/ana2/fja.scad | 13 | ||||
-rw-r--r-- | šola/ana2/kolokvij1.lyx | 976 | ||||
-rw-r--r-- | šola/ars/ol2.odt | bin | 0 -> 17332 bytes | |||
-rw-r--r-- | šola/ds2/kolokvij2.lyx | 3368 | ||||
-rw-r--r-- | šola/ds2/teor.lyx | 7560 | ||||
-rw-r--r-- | šola/krožek/05-17.odp | bin | 0 -> 17283 bytes | |||
-rw-r--r-- | šola/krožek/funkcije.odp | bin | 0 -> 532101 bytes | |||
-rw-r--r-- | šola/la/dn8/dokument.lyx | 1349 | ||||
-rw-r--r-- | šola/la/dn8/sage.png | bin | 0 -> 158070 bytes | |||
-rw-r--r-- | šola/la/kolokvij4.lyx | 1068 | ||||
-rw-r--r-- | šola/p2/dn/DN07a_63230317.c | 25 | ||||
-rw-r--r-- | šola/p2/dn/DN07b_63230317.c | 24 | ||||
-rw-r--r-- | šola/p2/dn/DN09b_63230317.c | 48 |
24 files changed, 15131 insertions, 6 deletions
diff --git a/skripti/365_prenesi_omejene_vsebine.sh b/skripti/365_prenesi_omejene_vsebine.sh index f9a8c69..16f527d 100755 --- a/skripti/365_prenesi_omejene_vsebine.sh +++ b/skripti/365_prenesi_omejene_vsebine.sh @@ -7,13 +7,51 @@ set -euo pipefail page=`curl --fail-with-body https://365.rtvslo.si/oddaja/dnevnik/92` # outputa preveč shita za set -x set -x -for id in `find -type f -size 7931523c | sed -E 's/^.*\[([0-9]*)\].*$/\1/g'` # repair broken downloads (pravice potekle) -do - yt-dlp --no-continue http://365.rtvslo.si/arhiv/oddaja/$id -done -rm -f Error\ \[*\].mp4 +if mkdir lock +then + echo $$ > lock/pid +else + if [ -d /proc/`cat lock/pid` ] + then + echo ANOTHER INSTANCE IS ALREADY RUNNING + echo if you are sure that this is not the case: + echo rm -r `pwd`/lock + exit 1 + fi +fi +echo $$ > lock/pid +p=`rev <<<$0 | cut -d/ -f1 | rev` +t=`mktemp -p "" $p.XXX` +trap "rm $t; rm -r lock" EXIT dnevnik_id=`grep href=./arhiv/dnevnik <<<"$page" | cut -d\" -f2 | cut -d/ -f4 | head -n1` -client_id=`grep client-id <<<"$page" | cut -d\" -f2 | head -n1` +client_id=`grep 'client-id="' <<<"$page" | head -n1 | sed -E 's/^.*client-id="([^"]*)".*$/\1/'` +find . -size -12M -type f -name '*.mp4' > $t +while read file +do # grep for specific audio codec ... if sample rate is 44100 and (tv broadcast is at 48000) and smaller than 12M, it's most likely the dummy/pravicepotekle video + samplerate=`ffprobe "$file" 2>&1 | grep 0x6134706D | grep -Eo '[0-9]+ Hz' | cut -d\ -f1` + id=`grep -Eo "\[[0-9]+\]\.mp" <<<"$file" | grep -o '[0-9]*'` + if [ $samplerate -eq 48000 ] + then + continue + fi + api_response=`curl --fail-with-body https://api.rtvslo.si/ava/getRecordingDrm/$id?client_id=$client_id` + if [ "`jq --raw-output .response.expirationDate <<<$api_response | head -c1`" = "3" ] || [ $((`date +%s`-86400*3-`date --date "$(jq --raw-output .response.expirationDate <<<$api_response)" +%s`)) -gt 0 ] + then + rm "$file" # sike, zgleda ne poteče! oziroma sike, je že potekel + continue + fi + if [ $samplerate -eq 44100 ] + then + yt-dlp --no-continue http://365.rtvslo.si/arhiv/oddaja/$id # retry fu*ked up downloads + if [ `ls *\[$id\].mp4 | wc -l` -gt 1 ] + then + rm "$file" # kdaj se zgodi, da RTV spremeni ime oddaje, tedaj imam staro ime, ki ima shranjen dummy posnetek 7,6 MiB, ki neprestano triggera reload ... + fi + continue + fi + echo -e "Subject: unknown samplerate in 365.sh\n\nVideo file: $file\nsamplerate: $samplerate\n" | sendmail root +done < $t +rm -f Error\ \[*\].mp4 if [ ! -f zadnji ] then echo $dnevnik_id > zadnji diff --git a/skripti/completion.bash b/skripti/completion.bash new file mode 100644 index 0000000..fbe834d --- /dev/null +++ b/skripti/completion.bash @@ -0,0 +1,6 @@ +# completion skript za vse stvari v /skripti/ +_service_sh_complete() { + [ $1 = $3 ] && { COMPREPLY=($(compgen -W "`service.sh list`" $2)); return; } + COMPREPLY=($(compgen -W "`service.sh $3 list`" $2)) +} +complete -F _service_sh_complete service.sh diff --git a/skripti/earhorn_m3u.sh b/skripti/earhorn_m3u.sh new file mode 100755 index 0000000..efffa1b --- /dev/null +++ b/skripti/earhorn_m3u.sh @@ -0,0 +1,43 @@ +#!/bin/bash +set -eo pipefail +# $1 je prefix +# $2 je prejšnji file (za caching dolžin) +# $3=1 forsira cache kljub version failu +# zadnjega vnosa iz cache m3uja nikdar ne bom upošteval, ker je lahko nedokončan file (no, v earhornu ne, ampak v stream_archive.sh) +echo "#EXTM3U" +ver="#earhorn_m3u.sh različica 0" +echo $ver +cache=0 +if [ ! x$3 = x ] || [ ! x$2 = x ] && [ -f $2 ] +then + if [ x"`head -n2 $2 | tail -n1`" = x"$ver" ] + then + cache=1 + fi +fi +find 2* -type f | while read file +do + date=`cut -d/ -f1,2,3 <<<$file | sed s,/,-,g` + time=`cut -d/ -f4 <<<$file | cut -d. -f1` + h=${time:0:2} + m=${time:2:2} + s=${time:4:2} + dol="" + if [ $cache -eq 1 ] + then # tu smatram, da vsaka datoteka zavzema 3 vrstice + head -n-3 $2 | grep -B2 ^$1$file$ && continue + fi # prvič v head ko rečem -n-3 in drugič v grep, ko rečem -B2 + set +e + dol=`ffprobe -v error -show_entries format=duration -of default=noprint_wrappers=1:nokey=1 $file` + koda=$? + set -e + if [ $koda -eq 0 ] + then + echo "#EXTINF:$dol,Radijski arhiv" + echo "#EXT-X-PROGRAM-DATE-TIME:${date}T$h:$m:${s}Z" + echo $1$file + else + echo "# NAPAKA $file ffprobe vrnil $koda" + echo "NAPAKA $file ffprobe vrnil $koda" >&2 + fi +done diff --git a/skripti/init_script_interpreter.sh b/skripti/init_script_interpreter.sh new file mode 100755 index 0000000..fc461ce --- /dev/null +++ b/skripti/init_script_interpreter.sh @@ -0,0 +1,115 @@ +#!/bin/bash +set -eo pipefail +name=`rev <<<$1 | cut -d/ -f1 | rev` +SVCNAME=$name +pidfile=/run/user/`id -u`/$name.pid +[ -w /run/user/`id -u`/log ] && stdout_logger="logger --socket /run/user/`id -u`/log" || stdout_logger=logger # failback to logging to syslog +stderr_logger=$stdout_logger +[ -r `sed s,/init.d/,/conf.d/, <<<$1` ] && . `sed s,/init.d/,/conf.d/, <<<$1` +. $1 +type eerror > /dev/null 2> /dev/null || eerror() { + echo NAPAKA: $@ +} +type ewarn > /dev/null 2> /dev/null || ewarn() { + echo OPOZORILO: $@ +} +type einfo > /dev/null 2> /dev/null || einfo () { + echo INFO: $@ +} +type ebegin > /dev/null 2> /dev/null || ebegin() { + echo ZAČENJAM: $@ +} +type need_net > /dev/null 2> /dev/null || need_net() { + [ `ip a | grep ^[[:digit:]] | wc -l` -gt 1 ] || { eerror "Ni omrežnih vmesnikov poleg lo." && return 1; } + [ `ip route | wc -l` -gt 1 ] || { eerror "Usmerjevalna tabela je prazna." && return 1; } +} +type need_clock > /dev/null 2> /dev/null || need_clock() { + year=`date +%Y` + [ $? -eq 0 ] || { eerror "Ura ne deluje." && return 1; } + [ $year -ge 2024 ] || { eerror "Ura je narobe nastavljena." && return 1; } +} +type need_hostname > /dev/null 2> /dev/null || need_hostname() { + hn=`hostname` + [ $? -eq 0 ] || { eerror "Ni mogoče pridobiti imena gostitelja." && return 1; } + [ x$hn = x ] && { eerror "Ni imena gostitelja" && return 1; } + [ x$hn = xlocalhost ] && { eerror "Ime gostitelja je localhost." && return 1; } + return 0 +} +needs= +needs_izpolnjen=0 +type need > /dev/null 2> /dev/null || need() { + for i in $@ + do + needs="$needs $i" + need_$i || { eerror "Odvisnost $i ni izpolnjena." && needs_izpolnjen=1; } + done +} +provides= +type provide > /dev/null 2> /dev/null || provide() { + for i in $@ + do + provides="$provides $i" + done +} +type depend > /dev/null 2> /dev/null || depend() { + return +} +type check_dependencies > /dev/null 2> /dev/null || check_dependencies() { + depend + return $needs_izpolnjen +} +type needs > /dev/null 2> /dev/null || needs() { + depend + echo $needs +} +type provides > /dev/null 2> /dev/null || provides() { + depend + echo $provides +} +type checkconfig > /dev/null 2> /dev/null || checkconfig() { + return +} +type start > /dev/null 2> /dev/null || start() { + checkconfig || { eerror "Checkconfig ni uspel." && return 1; }; + check_dependencies || { eerror "Odvisnosti niso izpolnjene."; return 1; } + ebegin "Zaganjam $SVCNAME" + start-stop-daemon --start --verbose `[ x$command_background = true ] || echo --background --make-pidfile` `[ x$chdir = x ] || echo --chdir $chdir` --stdout-logger "$stdout_logger" --stderr-logger "$stderr_logger" --exec $command --pidfile $pidfile -- $command_args || { eerror "start-stop-daemon ni uspel. Koda napake: $?"; return 1; } +} +type stop > /dev/null 2> /dev/null || stop() { + ebegin "Ustavljam $SVCNAME" + start-stop-daemon --stop --pidfile $pidfile --verbose --exec $command -- $command_args || { eerror "start-stop-daemon ni uspel. Koda napake: $?"; return 1; } +} +type restart > /dev/null 2> /dev/null || restart() { + checkconfig || { eerror "Checkconfig ni uspel." && return 1; }; + stop || ewarn "Stop ni uspel. Vseeno poskušam zagnati storitev." + start || { eerror "Start ni uspel." && return 1; } +} +type reload > /dev/null 2> /dev/null || reload() { + checkconfig || { eerror "Checkconfig ni uspel." && return 1; }; + ebegin "Pošiljam SIGHUP storitvi $SVCNAME" + start-stop-daemon --signal HUP $start_stop_daemon_args +} +type status > /dev/null 2> /dev/null || status() { + if [ -r $pidfile ] + then + if [ -d /proc/`cat $pidfile` ] + then + einfo started + else + einfo crashed + return 3 + fi + else + einfo stopped + return 3 + fi +} +type list > /dev/null 2> /dev/null || list() { + echo start stop restart $extra_commands $extra_started_commands $extra_stopped_commands status reload list check_dependencies needs provides +} +if type $2 > /dev/null 2> /dev/null +then + ${@:2} +else + eerror "Ta ukaz ne obstaja!" +fi diff --git a/skripti/rš_glasbena_oprema.py b/skripti/rš_glasbena_oprema.py new file mode 100644 index 0000000..8d53702 --- /dev/null +++ b/skripti/rš_glasbena_oprema.py @@ -0,0 +1,31 @@ +#!/usr/bin/python3 +import feedparser +import yt_dlp +from ollama import Client +from bs4 import BeautifulSoup +client = Client(host='http://splet.4a.si:80') +model = "llama2:13b-chat-fp16" +prompt = "The document below is text extracted from a Slovene radio station containing a tracklist. Extract the tracklist form the text below and output a CSV table in format \"artist,track name,album,duration,label\". Example output:\n\nThe Prodigy,Firestarter,The Fat of the Land,4:42,XL\nBJÖRK,LION SONG,,6:16," +def opreme(): + r = [] + for entry in feedparser.parse("https://radiostudent.si/taxonomy/term/589/*/feed").entries: + oprema = {"id": int(entry.id.split(" ")[0]), "title": entry.title, "link": entry.link, "published": entry.published_parsed, "authors": []} + for author in entry.authors: + oprema["authors"].append(author.name) + summary = BeautifulSoup(entry.summary, features="html.parser") + body = None + for i in summary.findAll("div"): + if "class" in i.attrs: + if "field-name-body" in i.attrs["class"]: + body = i + break + if "" + if body == None: + raise Exception("body is None in " + entry.link) + body = body.text.replace("\r", "") + while "\n\n" in body: + body = body.replace("\n\n", "\n") + r.append(oprema) + return r +if __name__ == "__main__": + opreme()
\ No newline at end of file diff --git a/skripti/service.sh b/skripti/service.sh new file mode 100755 index 0000000..2617253 --- /dev/null +++ b/skripti/service.sh @@ -0,0 +1,28 @@ +#!/bin/bash +[ x$INITPATH = x ] && for i in `tr : ' ' <<<$MORE_INITPATH` `tr : ' ' <<<$XDG_CONFIG_DIRS` ~/.config ~ /etc +do + [ -d $i/init.d ] && INITPATH=$INITPATH:$i/init.d +done +if [ x$1 = xlist ] +then + for i in `tr : ' ' <<<$INITPATH` + do + if [ -w $i ] + then + ls $i + else + for j in $i/* + do + [ ! -x $j ] && continue + head -n1 $j | grep "#!.*openrc-run" > /dev/null 2> /dev/null && continue + echo `rev <<<$j | cut -f/ -d1 | rev` + done + fi + done + exit 1 +fi +[ x$1 = x ] && { echo -e "Uporaba:\n\t$0 list\n\t$0 <storitev> <dejanje>"; exit 1; } +for i in `tr : ' ' <<<$INITPATH` +do + [ -x $i/$1 ] && exec $i/$@ +done diff --git a/skripti/stream_archive.sh b/skripti/stream_archive.sh new file mode 100755 index 0000000..b218246 --- /dev/null +++ b/skripti/stream_archive.sh @@ -0,0 +1,37 @@ +#!/bin/bash +# $1 naj bo icecast host kruljo.radiostudent.si +# $2 naj bo icecast http port 8000 +# $3 naj bo icecast endpoint /ehiq +# $4 naj bo končnica datotek .mp3 +# v CWD delam imenike in datoteke in sem glede tega kompatibilen z earhornom: +# Posnetek 2024/06/03/202136.mp3 se začne 2024-06-03T20:21:36 vedno UTC. +# Posnetkov ne splittam na uro. Posnetek se splitta le takrat, ko se zgodi napaka (stream crkne) -- takrat začnem pisati v novo datoteko +# požvižgam se na vsebino datotek. zame so to le bajti. v tem nisem earhornski. +# earhornski nisem tudi zato, ker se zadnja datoteka stalno veča, kar se ne dogaja pri earhornu. na to bodite pozorni. k sreči earhorn_m3u.sh ve za to obnašanje +set -xeuo pipefail +host=$1 +port=$2 +endpoint=$3 +kon=$4 +mistakes=0 +while : +do + filename=`date --utc +%Y/%m/%d/%H%m%S$kon` + mkdir -p `cut -d/ -f1-3 <<<$filename` + start=$SECONDS + set +e + nc $host $port <<<"GET $endpoint HTTP/1.0"$'\r\n\r' > $filename + koda=$? + set -e + echo TCP PREKINJEN! ZAČENJAM NOV POSNETEK! IZHODNA KODA nc je $koda, datum je `date` + if [ $(($SECONDS-$start)) -lt 300 ] + then + mistakes=$(($mistakes+1)) + if [ $mistakes -gt 3 ] + then + sleep $((2**($mistakes-3))) + fi + else + mistakes=0 + fi +done diff --git a/skripti/videonadzor.sh b/skripti/videonadzor.sh new file mode 100755 index 0000000..e097c58 --- /dev/null +++ b/skripti/videonadzor.sh @@ -0,0 +1,100 @@ +#!/bin/bash +# SKRIPT za snemanje kamere, katere firmware je example arduino sketch za ESP32CAM CameraWebServer +# Neprestano zahteva JPEGe na /capture endpointu +# Vsako minuto iz JPEGov naredi MKV in nato JPEGe izbriše +# Vsako uro naredi iz MKVjev MP4 in nato MKVje izbriše +set -uo pipefail +q=4 +since=0 +rm -f concat.txt +curl --no-progress-meter "$1/control?var=framesize&val=13" +while : +do + curl --no-progress-meter "$1/control?var=quality&val=$q" + t=`date --utc --iso-8601=ns | cut -d+ -f1` + oldstart=`ls --sort=time | grep jpeg$ | tail -n1 | cut -d: -f1,2` + if [ ! $oldstart = `cut -d: -f1,2 <<<"$t"` ] && [ ! -f concat.txt ] + then + echo "ffconcat version 1.0" > concat.txt + prev=devica + first=ERROR + for i in $oldstart:*.jpeg konec + do + if [ $prev = devica ] + then + first=`rev <<<"$i" | cut -d. -f2- | rev` + prev=$i + continue + fi + echo "file 'file:$prev'" >> concat.txt + if [ ! $i = konec ] + then + echo "duration 0`dc <<<"10k$(date --utc --date $(rev <<<"$i" | cut -d. -f2- | rev) +%s.%N) $(date --utc --date $(rev <<<"$prev" | cut -d. -f2- | rev) +%s.%N)"-p`" >> concat.txt + else + echo "duration 1" >> concat.txt + fi + prev=$i + done + { + ffmpeg -f concat -safe 0 -i concat.txt -vsync vfr file:$first.mkv + if [ -s $first.mkv ] + then + while read line + do + grep ^file <<<$line > /dev/null && rm `cut -d: -f2- <<<$line | cut -d\' -f1` + done < concat.txt + fi + oldstart=`ls --sort=time | grep mkv$ | tail -n1 | cut -d: -f1` + if [ ! $oldstart = `cut -d: -f1 <<<"$t"` ] + then + echo "ffconcat version 1.0" > concat.txt + prev=devica + for i in $oldstart:*.mkv konec + do + if [ $prev = devica ] + then + prev=$i + continue + fi + echo "file 'file:$prev'" >> concat.txt + if [ ! $i = konec ] + then + echo "duration 0`dc <<<"10k$(date --utc --date $(rev <<<"$i" | cut -d. -f2- | rev) +%s.%N) $(date --utc --date $(rev <<<"$prev" | cut -d. -f2- | rev) +%s.%N)"-p`" >> concat.txt + fi + prev=$i + done + ffmpeg -f concat -safe 0 -i concat.txt -vsync vfr file:$oldstart.mp4 + if [ -s $oldstart.mp4 ] + then + while read line + do + grep ^file <<<$line > /dev/null && rm `cut -d: -f2- <<<$line | cut -d\' -f1` + done < concat.txt + fi + fi + rm concat.txt + } & + fi + curl --no-progress-meter --fail -o$t.jpeg $1/capture + curlexit=$? + if [ $curlexit -eq 22 ] + then + q=$(($q+1)) + echo "Setting quality to $q" + continue + fi + if [ ! $curlexit -eq 0 ] + then + echo ERROR!!! Curl returned with $curlexit + continue + fi + if [ $since -ge 1024 ] && [ $q -gt 4 ] + then + q=$(($q-1)) + fi + if [ ! `file $t.jpeg | grep -o [1-9][0-9]*x[1-9][0-9]*` = 1600x1200 ] + then + echo Popravljam ločljivost. + curl --no-progress-meter "$1/control?var=framesize&val=13" + fi +done diff --git a/skripti/zone_axfr.py b/skripti/zone_axfr.py new file mode 100755 index 0000000..81460a6 --- /dev/null +++ b/skripti/zone_axfr.py @@ -0,0 +1,86 @@ +#!/usr/bin/python3 +import dns.zone +import dns.resolver +import json +import sys +domena = sys.argv[1] +strežniki = [dns.resolver.resolve(domena, "SOA")[0].mname] +for i in dns.resolver.resolve(domena, "NS"): + strežniki.append(i.target) +naslovi = [] +for strežnik in strežniki: + for i in dns.resolver.resolve(strežnik, "AAAA"): + naslovi.append(i.address) + for i in dns.resolver.resolve(strežnik, "A"): + naslovi.append(i.address) +for naslov in naslovi: # opcijsko dodaj tule kakšen try catch + zone = None + zone = dns.zone.from_xfr(dns.query.xfr(naslov, domena)) + if zone != None: + break +config = None +try: + config = json.loads(b''.join(zone["_urejevalnik"].get_rdataset(dns.rdataclass.IN, dns.rdatatype.TXT)[0].strings).decode()) +except KeyError: + pass +except json.decoder.JSONDecodeError: + pass +if config == None: + berime = """; Dobrodošli v preprost urejevalnik DNS zapisov. +; Komentarji se shranijo v DNS strežnik in so javni. Morajo biti na samostojnih vrsticah. +; Te komentarje z navodili lahko izbrišete -- ne bodo se ponovno pojavili. +; Nove zapise naložite na strežnik z ukazom zone_update.py zonefile.db +; Zapise prenesete iz strežnika z ukazom zone_axfr.py domena > zonefile.db +; Prva vrstica je konfiguracijski zapis v JSON obliki. Naslednje podatke lahko spremenite: +; "t": privzeti TTL, ki se uporabi, če zapis v datoteki nima TTLja +; "+": koliko naj prištejem serijski številki pred nalaganjem na strežnik + +""" + config = {"v": 0, "d": domena, "c": {"@ SOA": berime}, "t": 1, "+": 100, "i": {}} +configout = config.copy() +del configout["c"] +del configout["i"] +print(f"{json.dumps(configout)}") +for r in zone.iterate_rdatas(): + if r[0].to_unicode() == "_urejevalnik" or r[2].rdtype in [dns.rdatatype.RRSIG, dns.rdatatype.NSEC, dns.rdatatype.NSEC3, dns.rdatatype.DNSKEY]: + continue + commentkey = r[0].to_unicode() + " " + r[2].rdtype.name + if commentkey in config["c"].keys(): + print(config["c"][commentkey], end="") + del config["c"][commentkey] + konec = "\t" + if r[0].to_unicode() in config["i"].keys(): + konec = config["i"][r[0].to_unicode()] + print(r[0].to_unicode(), end=konec) + if r[1] != config["t"]: + print(r[1], end="") + print("\t", end="") + if r[2].rdclass != dns.rdataclass.IN: + print(r[2].rdataclass.name, end="\t") + print(r[2].rdtype.name, end="\t") + if r[2].rdtype == dns.rdatatype.TXT: + prvič = True + for string in r[2].strings: + if prvič: + prvič = False + else: + print(" ", end="") + bajti = b'' + for char in string: + if char < ord(b' '): + bajti += b'\\' + ("%03d" % ord(char)).encode() + else: + bajti += bytes([char]) + niz = "" + for znak in bajti.replace(b'\\', b'\\\\').replace(b'"', b'\\"').decode('utf-8', errors='surrogateescape'): + if '\udc80' <= znak <= '\udcff': + niz += '\\'+("%03d" % (ord(znak)-0xdc00)) + else: + niz += znak + + print('"' + niz + '"', end="") + else: + print(r[2].to_text(), end="") + print() +for i in config["c"].items(): + print(i[1], end="") diff --git a/skripti/zone_update.py b/skripti/zone_update.py new file mode 100755 index 0000000..ead2c39 --- /dev/null +++ b/skripti/zone_update.py @@ -0,0 +1,174 @@ +#!/usr/bin/python3 +import dns.zone +import dns.resolver +import dns.update +import dns.tsigkeyring +import json +import sys +import math +with open(sys.argv[1], "r") as db: + lines = db.readlines() +newconfig = json.loads(lines.pop(0)[:-1]) +domena = newconfig["d"] +strežniki = [dns.resolver.resolve(domena, "SOA")[0].mname] +naslovi = [] +for strežnik in strežniki: + for i in dns.resolver.resolve(strežnik, "AAAA"): + naslovi.append(i.address) + for i in dns.resolver.resolve(strežnik, "A"): + naslovi.append(i.address) +for naslov in naslovi: # opcijsko dodaj tule kakšen try catch + zone = None + zone = dns.zone.from_xfr(dns.query.xfr(naslov, domena)) + if zone != None: + break +config = None +try: + config = json.loads(b''.join(zone["_urejevalnik"].get_rdataset(dns.rdataclass.IN, dns.rdatatype.TXT)[0].strings).decode()) +except KeyError: + pass +except json.decoder.JSONDecodeError: + pass +if config == None: + config = {"v": 0, "d": domena, "c": {}, "t": 1, "+": 100, "i": {}} +rrs = [] +for r in zone.iterate_rdatas(): + if r[0].to_unicode() == "_urejevalnik" or r[2].rdtype in [dns.rdatatype.RRSIG, dns.rdatatype.NSEC, dns.rdatatype.NSEC3, dns.rdatatype.DNSKEY]: + continue + commentkey = r[0].to_unicode() + " " + r[2].rdtype.name + komentar = "" + if commentkey in config["c"].keys(): + komentar = config["c"][commentkey] + del config["c"][commentkey] + konec = "\t" + if r[0].to_unicode() in config["i"].keys(): + konec = config["i"][r[0].to_unicode()] + vrednost = "" + if r[2].rdtype == dns.rdatatype.TXT: + for string in r[2].strings: + bajti = b'' + for char in string: + if char < ord(b' '): + bajti += b'\\' + ("%03d" % ord(char)).encode() + else: + bajti += bytes([char]) + niz = "" + for znak in bajti.replace(b'\\', b'\\\\').replace(b'"', b'\\"').decode('utf-8', errors='surrogateescape'): + if '\udc80' <= znak <= '\udcff': + niz += '\\'+("%03d" % (ord(znak)-0xdc00)) + else: + niz += znak + vrednost = '"' + niz + '"' + else: + vrednost = r[2].to_text() + rrs.append((r[1], r[0].to_unicode(), komentar, None, r[2].rdclass, r[2].rdtype, vrednost)) +komentar = "" +lineno = 1 +novikomentarji = {} +novikonci = {} +keyring = None +plus = 0 +minus = 0 +if len(sys.argv) == 3: + with open(sys.argv[2]) as file: + ključ = file.read() + keyring = dns.tsigkeyring.from_text({ključ.split()[ključ.split().index("key")+1].replace('"', "").replace(";", ""): ključ.split()[ključ.split().index("secret")+1].replace('"', "").replace(";", "")}) +update = dns.update.Update(domena, keyring=keyring) +while True: + try: + lineno += 1 + line = lines.pop(0)[:-1] # odstranimo zadnji \n, ki ga zraven da .readlines + if len(line.split()) == 0 or line[0] == ';' or line[0] == '#' or line[0] == '/': + komentar += line + "\n" + continue + ime = line.split()[0] + konec = "" + index = len(ime) + while line[index] in ["\t", " "]: + konec += line[index] + index += 1 + nizi = line.split()[1:] + tip = None + razred = None + ttl = None + while tip == None: + try: + ttl = int(nizi[0]) + nizi.pop(0) + except ValueError: + pass + try: + razred = dns.rdataclass.from_text(nizi[0]) + nizi.pop(0) + except dns.rdataclass.UnknownRdataclass: + pass + try: + tip = dns.rdatatype.from_text(nizi[0]) + for i in [" ", "\t"]: + for j in [" ", "\t"]: + try: + datastart = line.index(i+nizi[0]+j)+len(i+nizi[0]+i) + except ValueError: + continue + break + else: + continue + break + nizi.pop(0) + except dns.rdatatype.UnknownRdatatype: + pass + if tip == None: + print(f"NAPAKA: na vrstici {lineno} ne najdem tipa zapisa. Vrstica je lahko bodisi komentar, ki se začne z ';', bodisi je v obliki IME [TTL={newconfig['t']}] [CLASS=IN] TIP PODATKI.") + print(f"Vsebina neveljavne vrstice: " + line) + sys.exit(1) + while line[datastart] in [" ", "\t"]: + datastart += 1 + data = line[datastart:] + if razred == None: + razred = dns.rdataclass.IN + if ttl == None: + ttl = newconfig["t"] + ime = dns.name.from_unicode(ime, dns.name.from_unicode(domena)).choose_relativity(dns.name.from_unicode(domena), True).to_unicode() + if tip == dns.rdatatype.SOA: + data = data.split() + data[2] = str(int(data[2])+newconfig["+"]) + data = " ".join(data) + tapl = (ttl, ime, komentar, None, razred, tip, data) + if komentar != "": + novikomentarji[ime + " " + tip.to_text(tip)] = komentar + if konec != "\t": + novikonci[ime] = konec + if not tapl in rrs: + print("+ " + komentar.replace("\n", "\n+ ") + ime + konec + str(ttl) + "\t" + razred.to_text(razred) + "\t" + tip.to_text(tip) + "\t" + data) + plus += 1 + update.add(ime, ttl, tip, data) + else: + rrs.remove(tapl) + komentar = "" + except IndexError: + break +obstoječ = "" # zadnji komentar +for komentar in config["c"].values(): + obstoječ += komentar +for rr in rrs: + print("- " + komentar.replace("\n", "\n- ") + rr[1] + konec + str(rr[0]) + "\t" + rr[4].to_text(rr[4]) + "\t" + rr[5].to_text(rr[5]) + "\t" + rr[6]) + minus += 1 + update.delete(rr[1], rr[5].to_text(rr[5]), rr[6]) +if obstoječ != komentar: + print("- " + "\n+ ".join(obstoječ.split("\n"))) + print("+ " + "\n+ ".join(komentar.split("\n"))) + plus += 1 + minus += 1 +novikomentarji["z"] = komentar +newconfig["c"] = novikomentarji +newconfig["i"] = novikonci +odziv = input(f"(-{minus}/+{plus}) Ali želite te spremembe poslati na strežnik? [Y/D/J/n] ") +if len(odziv) != 0 and odziv[0] in ["n", "N", "0", "f", "F"]: + print("Prekinjam. Nasvidenje!") + sys.exit(0) +jason = json.dumps(newconfig) +jasonsplit = " ".join(['"' + jason[i*255:i*255+255].replace("\\", "\\\\").replace('"', '\\"') + '"' for i in range(math.ceil(len(jason)/255))]) +update.replace("_urejevalnik", 1, dns.rdatatype.TXT, jasonsplit) +response = dns.query.tcp(update, naslov) +print("Poslal zahtevo. Odziv strežnika:") +print(response) diff --git a/skripti/ž b/skripti/ž new file mode 100644 index 0000000..d18de36 --- /dev/null +++ b/skripti/ž @@ -0,0 +1,36 @@ +#!/bin/bash +# $1 naj bo icecast host kruljo.radiostudent.si +# $2 naj bo icecast http port 8000 +# $3 naj bo icecast endpoint /ehiq +# $4 naj bo končnica datotek .mp3 +# v CWD delam imenike in datoteke in sem glede tega kompatibilen z earhornom: +# Posnetek 2024/06/03/202136.mp3 se začne 2024-06-03T20:21:36 vedno UTC. +# Posnetkov ne splittam na uro. Posnetek se splitta le takrat, ko se zgodi napaka (stream crkne) -- takrat začnem pisati v novo datoteko +# požvižgam se na vsebino datotek. zame so to le bajti. v tem nisem earhornski. +host=$1 +port=$2 +endpoint=$3 +kon=$4 +mistakes=0 +set -xeuo pipefail +while : +do + filename=`date --utc +%Y/%m/%d/%H%m%S$kon` + mkdir -p `cut -d/ -f1-3 <<<$filename` + start=$SECONDS + set +e + nc $host $port <<<"GET $endpoint HTTP/1.0"$'\r\n\r' > $filename + koda=$? + set -e + echo TCP PREKINJEN! ZAČENJAM NOV POSNETEK! IZHODNA KODA nc je $koda, datum je `date` + if [ $(($SECONDS-$start)) -lt 300 ] + then + mistakes=$(($mistakes+1)) + if [ $mistakes -gt 3 ] + then + sleep $((2**($mistakes-3))) + fi + else + mistakes=0 + fi +done diff --git a/šola/ana2/fja.scad b/šola/ana2/fja.scad new file mode 100644 index 0000000..4d18a17 --- /dev/null +++ b/šola/ana2/fja.scad @@ -0,0 +1,13 @@ +echo(version=version()); +function sinh(x) = (exp(1)-exp(-x))/2; +function f(x, y, z) = sinh(z)*sinh(y)-sin(x); +epsilon = 0.01; +rob = 1; +korak = 0.1; +particle = 0.1; +for (x = [-rob : korak : rob]) + for (y = [-rob : korak : rob]) + for (z = [-rob : korak : rob]) + if (f(x, y, z) < epsilon) + translate([x, y, z]) + cube(particle);
\ No newline at end of file diff --git a/šola/ana2/kolokvij1.lyx b/šola/ana2/kolokvij1.lyx new file mode 100644 index 0000000..a4f9569 --- /dev/null +++ b/šola/ana2/kolokvij1.lyx @@ -0,0 +1,976 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\begin_preamble +\usepackage{siunitx} +\usepackage{pgfplots} +\usepackage{listings} +\usepackage{multicol} +\usepackage{amsmath} +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}} +\DeclareMathOperator{\g}{g} +\DeclareMathOperator{\sled}{sled} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Cir}{Cir} +\DeclareMathOperator{\ecc}{ecc} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\diam}{diam} +\newcommand\euler{e} +\end_preamble +\use_default_options true +\begin_modules +enumitem +\end_modules +\maintain_unincluded_children false +\language slovene +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics xetex +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification false +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 1cm +\topmargin 1cm +\rightmargin 1cm +\bottommargin 2cm +\headheight 1cm +\headsep 1cm +\footskip 1cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style german +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +setlength{ +\backslash +columnseprule}{0.2pt} +\backslash +begin{multicols}{2} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Podmnožice v evklidskih prostorih +\end_layout + +\begin_layout Standard +\begin_inset Formula $A$ +\end_inset + + zaprta, če +\begin_inset Formula $\forall$ +\end_inset + + zaporedje s členi v +\begin_inset Formula $A:$ +\end_inset + + vsa stekališča v +\begin_inset Formula $A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $A$ +\end_inset + + kompaktna, če +\begin_inset Formula $\forall$ +\end_inset + + zaporedje s členi v +\begin_inset Formula $A$ +\end_inset + +: +\begin_inset Formula $\exists$ +\end_inset + + stekališče v +\begin_inset Formula $A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $f$ +\end_inset + + je kompozitum zveznih +\begin_inset Formula $\Rightarrow$ +\end_inset + + +\begin_inset Formula $f$ +\end_inset + + zvezna +\end_layout + +\begin_layout Standard +za +\begin_inset Formula $x\in\mathbb{R}^{k}$ +\end_inset + +: +\begin_inset Formula $\text{\left|\left|x\right|\right|\ensuremath{\coloneqq}}\sqrt{x_{1}^{2}+\cdots+x_{k}^{2}}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $A\subseteq\mathbb{R}^{k}$ +\end_inset + + omejena +\begin_inset Formula $\Leftrightarrow\exists M\in\mathbb{R}\forall x\in A:\left|\left|x\right|\right|<M$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $K\left(s\in\mathbb{R}^{k},M\in\mathbb{R}\right)\coloneqq\left\{ s+x\in\mathbb{R}^{k};\text{\left|\left|x\right|\right|}<M\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $x\in A$ +\end_inset + + notranja +\begin_inset Formula $\Leftrightarrow\exists\varepsilon\ni:K\left(x,\varepsilon\right)\subset A$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $A\subseteq\mathbb{R}^{k}$ +\end_inset + + odprta +\begin_inset Formula $\Leftrightarrow\forall x\in A:x$ +\end_inset + + notranja +\end_layout + +\begin_layout Standard +\begin_inset Formula $A\subseteq\mathbb{R}^{k}$ +\end_inset + + zaprta +\begin_inset Formula $\Leftrightarrow\mathbb{R}^{k}\setminus A$ +\end_inset + + odprta +\end_layout + +\begin_layout Standard +\begin_inset Formula $x\in\mathbb{R}^{k}$ +\end_inset + + stekališče +\begin_inset Formula $A\Leftrightarrow\forall\varepsilon>0:K\left(x,\varepsilon\right)\cup\left(A\setminus\left\{ x\right\} \right)\not=\emptyset$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $x\in\mathbb{R}^{k}$ +\end_inset + + izolirana točka +\begin_inset Formula $A\Leftrightarrow x$ +\end_inset + + ni stekališče +\begin_inset Formula $A$ +\end_inset + + +\end_layout + +\begin_layout Standard +Zaporedje +\begin_inset Formula $a_{n}:\mathbb{N}\to\mathbb{R}^{k}$ +\end_inset + + konvergira proti +\begin_inset Formula $a\in\mathbb{R}^{k}$ +\end_inset + + kadar +\begin_inset Formula $\forall\varepsilon>0\exists N\in\mathbb{N}\forall n>N:\left|\left|a_{n}-a\right|\right|<\varepsilon$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $s\in\mathbb{R}^{k}$ +\end_inset + + stekališče zap. + +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + v +\begin_inset Formula $\text{\ensuremath{\varepsilon}}-$ +\end_inset + +okolici +\begin_inset Formula $s$ +\end_inset + + je +\begin_inset Formula $\infty$ +\end_inset + +mnogo členov +\end_layout + +\begin_layout Standard +Vsako omejeno zaporedje ima stekališče. +\end_layout + +\begin_layout Section +Funkcije več spremenljivk +\end_layout + +\begin_layout Standard +fja +\begin_inset Formula $k$ +\end_inset + + spremenljivk je preslikava +\begin_inset Formula $f:D\subseteq\mathbb{R}^{k}\to\mathbb{R}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $L\in\mathbb{R}$ +\end_inset + + je limita +\begin_inset Formula $f:D\subseteq\text{\ensuremath{\mathbb{R}^{k}}}\to\mathbb{R}$ +\end_inset + + v stekališču +\begin_inset Formula $a\in D$ +\end_inset + +, če +\begin_inset Formula $\forall\varepsilon>0\exists\delta>0\forall x\in D\setminus\left\{ a\right\} :\left|\left|x-a\right|\right|<\delta\Rightarrow\left|f\left(x\right)-L\right|<\varepsilon$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\lim_{x\to a}\left(f\oslash g\right)x=\lim_{x\to a}fx\oslash\lim_{x\to a}gx$ +\end_inset + +, če obstajata. + +\begin_inset Formula $\oslash\in\left\{ +,-,\cdot\right\} $ +\end_inset + +. + +\begin_inset Formula $\oslash$ +\end_inset + + je lahko deljenje, kadar +\begin_inset Formula $\lim_{x\to a}gx\not=0$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $f$ +\end_inset + + zvezna v +\begin_inset Formula $a\Leftrightarrow\forall\varepsilon\exists\delta\forall x\in D:\left|\left|x-a\right|\right|<\delta\Rightarrow\left|fx-fa\right|<\varepsilon$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $f$ +\end_inset + + zvezna +\begin_inset Formula $\Leftrightarrow\forall a\in D:f$ +\end_inset + + zvezna v +\begin_inset Formula $a$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $f$ +\end_inset + + zvezna v stekališču +\begin_inset Formula $a\Leftrightarrow\lim_{x\to a}fx=fa$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $A\subseteq\mathbb{R}^{k}$ +\end_inset + + kompaktna, +\begin_inset Formula $f:A\to\mathbb{R}$ +\end_inset + + zvezna +\begin_inset Formula $\Rightarrow f$ +\end_inset + + omejena in doseže maksimum in minimum (obstoj globalnega ekstrema). +\end_layout + +\begin_layout Standard +\begin_inset Formula $f,g$ +\end_inset + + zv. + +\begin_inset Formula $\Rightarrow f\oslash g$ +\end_inset + + zv. + +\begin_inset Formula $\oslash\in\left\{ +,-,\cdot,\circ\right\} ,\oslash=/\Leftrightarrow\forall x:gx\not=0$ +\end_inset + + +\end_layout + +\begin_layout Section +Odvodi funkcij več spremenljivk +\end_layout + +\begin_layout Standard +\begin_inset Formula $f_{x_{i}}a$ +\end_inset + +, +\begin_inset Formula $i\in\left\{ 1..k\right\} $ +\end_inset + + je odvod fje +\begin_inset Formula $x_{i}\to f\left(a_{1},\dots,a_{i-1},x_{i},a_{i+1},\dots,a_{k}\right)$ +\end_inset + + v točki +\begin_inset Formula $a_{i}$ +\end_inset + +. + +\begin_inset Formula $f_{x_{i}}a=\lim_{x_{i}\to a_{i}}\frac{f\left(a_{1},\dots,a_{i-1},x_{i},a_{i+1},\dots,a_{k}\right)-fa}{x_{i}-a_{i}}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Tang. + ravn. + v +\begin_inset Formula $a=\left(b,c\right)$ +\end_inset + + +\begin_inset Formula $a=\left(b,c\right)$ +\end_inset + +: +\begin_inset Formula $z=fa+f_{x}a\cdot\left(x-b\right)+f_{y}a\cdot\left(y-c\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $f$ +\end_inset + + je odvedljiva v +\begin_inset Formula $a\Leftrightarrow\lim_{h\to\left(0,0\right)}\frac{R_{a}\left(a+h\right)}{\left|\left|h\right|\right|}=0$ +\end_inset + +, kjer je +\begin_inset Formula $R_{a}\left(a+h\right)\coloneqq f\left(a+h\right)-fa-f_{x}\left(a\right)\cdot u+f_{y}\left(a\right)\cdot v$ +\end_inset + + za +\begin_inset Formula $h=\left(u,v\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $dfa\coloneqq\left[f_{x_{1}}a\cdots f_{x_{k}}a\right]=\nabla fa,dfa\cdot h\coloneqq f_{x_{1}}a\cdot h_{1}+\cdots+f_{x_{k}}a\cdot h_{k}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $f$ +\end_inset + + v +\begin_inset Formula $a$ +\end_inset + + odvedljiva +\begin_inset Formula $\Rightarrow f$ +\end_inset + + v +\begin_inset Formula $a$ +\end_inset + + zvezna +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall i\in\left\{ 1..k\right\} :\exists f_{x_{i}}\wedge f_{x_{i}}$ +\end_inset + + zvezna +\begin_inset Formula $\Rightarrow f$ +\end_inset + + odvedljiva +\end_layout + +\begin_layout Standard +Lagrange: +\begin_inset Formula $fx_{1}-fx_{2}=f'\xi\left(x_{2}-x_{1}\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $f\in C^{r}U\sim$ +\end_inset + + +\begin_inset Formula $f$ +\end_inset + + +\begin_inset Formula $r-$ +\end_inset + +krat zvezno odvedljiva v vsaki točki +\begin_inset Formula $U$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $f:U\subseteq\mathbb{R}^{k}\to\mathbb{R},f\in C^{2}U\Rightarrow\forall i,j\in\left\{ 1..k\right\} :f_{x_{i}}=f_{x_{j}}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\frac{df}{d\vec{s}}\left(x,y\right)=\lim_{t\to0}\frac{f((x,y)+t\vec{s})-f(x,y)}{t}=s_{1}f(x,y)+s_{2}f_{x}(x,y)$ +\end_inset + + +\end_layout + +\begin_layout Standard +Tangentna ravnina v +\begin_inset Formula $\left(x_{0},y_{0},f\left(x_{0},y_{0}\right)\right)$ +\end_inset + + je +\begin_inset Formula $f_{x}\left(x_{0},y_{0}\right)\left(x-x_{0}\right)+f_{y}\left(x_{0},y_{0}\right)\left(y-y_{0}\right)-z+f\left(x_{0},y_{0}\right)$ +\end_inset + + in razpenjata jo vektorja +\begin_inset Formula $\left(1,0,f_{x}\right)$ +\end_inset + + in +\begin_inset Formula $\left(0,1,f_{y}\right)$ +\end_inset + +. +\end_layout + +\begin_layout Section +Taylorjeva formula in verižno pravilo +\end_layout + +\begin_layout Standard +\begin_inset Formula $f$ +\end_inset + + diferenciabilna v +\begin_inset Formula $a\in\mathbb{R}^{k}\Rightarrow f\left(a+h\right)\cong fa+dfa\cdot h=fa+f_{x_{1}}a\cdot h_{1}+\cdots+f_{x_{k}}a\cdot h_{k}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $U\subseteq R^{k}$ +\end_inset + + odprta in +\begin_inset Formula $f:U\to\mathbb{R},f\in C^{n+1}U$ +\end_inset + +. + Naj bo +\begin_inset Formula $D_{f,r,a}$ +\end_inset + + vektor vseh parcialnih odvodov reda +\begin_inset Formula $r$ +\end_inset + + v točki +\begin_inset Formula $a$ +\end_inset + +. + Primer: +\begin_inset Formula $D_{f,2,a}=\left(f_{xx}a,2f_{xy}a+f_{yy}a\right)$ +\end_inset + +. + +\begin_inset Formula $D_{f,0,a}\coloneqq f\left(a\right)$ +\end_inset + +. + Naj bo +\begin_inset Formula $H_{r}$ +\end_inset + + vektor z vsemi kombinacijami dolžine +\begin_inset Formula $r$ +\end_inset + + komponent +\begin_inset Formula $h$ +\end_inset + +. + Primer: +\begin_inset Formula $H_{2}=\left(uu,2uv,vv\right)$ +\end_inset + +. + +\begin_inset Formula $H_{0}=1$ +\end_inset + +. + +\begin_inset Formula $D_{f,r,a}\cdot H_{r}$ +\end_inset + + je njun skalarni produkt. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +T_{f,a,n}\left(h_{1}=x-a,h_{2}=y-b\right)=\sum_{i=0}^{n}\frac{1}{i!}\left(D_{f,i,a}\cdot H_{i}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Section +Ekstremalni problemi +\end_layout + +\begin_layout Standard +Kandidati so +\begin_inset Formula $a$ +\end_inset + +, da +\begin_inset Formula $\nabla fa=0$ +\end_inset + + ali +\begin_inset Formula $f$ +\end_inset + + ni odv. + v +\begin_inset Formula $a$ +\end_inset + + ali +\begin_inset Formula $a$ +\end_inset + + robna točka. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +H\left(a,b\right)=\left[\begin{array}{cc} +f_{xx}\left(a,b\right) & f_{xy}\left(a,b\right)\\ +f_{yx}\left(a,b\right) & f_{yy}\left(a,b\right) +\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\det H\left(a,b\right)>0$ +\end_inset + +: +\begin_inset Formula $f_{xx}\left(a,b\right)>0$ +\end_inset + + l. + min., +\begin_inset Formula $f_{xx}\left(a,b\right)<0$ +\end_inset + + l. + max. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\det H\left(a,b\right)<0$ +\end_inset + + sedlo +\end_layout + +\begin_layout Standard +Izrek o implicitni funkciji: +\begin_inset Formula $D\subseteq\mathbb{R}^{2}$ +\end_inset + + odprta, +\begin_inset Formula $f:D\to\mathbb{R}$ +\end_inset + + zvezno parcialno odvedljiva. + +\begin_inset Formula $K=\left\{ \left(x,y\right)\in D;f\left(x,y\right)=0\right\} $ +\end_inset + +. + Za +\begin_inset Formula $\left(a,b\right)\in D$ +\end_inset + +, +\begin_inset Formula $f\left(a,b\right)=0\wedge\nabla f\left(a,b\right)\not=0\exists h\left(a\right)=b,f\left(x,h\left(x\right)\right)=0\forall x\in U$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Vezani ekstrem: +\begin_inset Formula $D^{\text{odp.}}\subseteq\mathbb{R}^{n}$ +\end_inset + +, +\begin_inset Formula $f:D\to\mathbb{R}$ +\end_inset + + diferenciabilna na +\begin_inset Formula $D$ +\end_inset + +. + let +\begin_inset Formula $g:D\to\mathbb{R}$ +\end_inset + + zvezno parcialno odvedljiva, +\begin_inset Formula $A\coloneqq\left\{ x\in D;gx=0\right\} $ +\end_inset + +. + +\begin_inset Formula $\exists$ +\end_inset + + vezani ekstrem +\begin_inset Formula $f$ +\end_inset + + pri pogoju +\begin_inset Formula $g\Leftrightarrow\nabla fa=\lambda\nabla ga$ +\end_inset + +. + Kandidati za vezane ekstreme so stac. + točke fje +\begin_inset Formula $F\left(x,\lambda\right)=fx-\lambda gx$ +\end_inset + +. +\end_layout + +\begin_layout Section +Krivulje in ploskve +\end_layout + +\begin_layout Standard +Pot v +\begin_inset Formula $\mathbb{R}^{3}\sim\vec{r}:I\to\mathbb{R}^{3},I\in\mathbb{R}$ +\end_inset + + interval. + +\begin_inset Formula $\forall t\in I:\vec{r}t=\left(xt,yt,zt\right)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Odvod poti: +\begin_inset Formula $\dot{\vec{r}}\left(t\right)=\left(\dot{x}\left(t\right),\dot{y}\left(t\right),\dot{z}\left(t\right)\right)$ +\end_inset + +. + +\begin_inset Formula $\dot{\vec{r}}t$ +\end_inset + + je tangentni vektor na krivuljo v točki +\begin_inset Formula $\vec{r}t$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Dolžina poti +\begin_inset Formula $\vec{r}:\left[a,b\right]\to\mathbb{R}^{3}$ +\end_inset + + je +\begin_inset Formula +\[ +L=\int_{a}^{b}\left|\dot{\vec{r}}t\right|dt=\int_{a}^{b}\sqrt{\dot{x}^{2}t+\dot{y}^{2}t}dt +\] + +\end_inset + +. +\end_layout + +\begin_layout Standard +Ploščina območja, ki ga omejuje krivulja, če je parametrizacija taka, da + je krivulja levo od +\begin_inset Formula $\vec{r}t$ +\end_inset + +: +\begin_inset Formula $\text{Pl\left(D\right)=\ensuremath{\frac{1}{2}\int_{a}^{b}\left(xt\dot{y}t-\dot{x}tyt\right)dt}}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Ploskev eksplicitno kot graf +\begin_inset Formula $f:D^{\text{odp.}}\subseteq\mathbb{R}^{2}\to\mathbb{R}$ +\end_inset + +. + +\begin_inset Formula $f$ +\end_inset + + difer. + v +\begin_inset Formula $\left(x,y\right)\Rightarrow$ +\end_inset + + v +\begin_inset Formula $\left(x,y,f\left(x,y\right)\right)$ +\end_inset + + definiramo tangentno ravnino z normalo +\begin_inset Formula $\left(-f_{x}\left(x,y\right),-f_{y}\left(x,y\right),1\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +Ploskev implicitno: +\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$ +\end_inset + + zvezno parcialno odvedljiva. +\begin_inset Formula +\[ +P=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3};f\left(x,y,z\right)=0\right\} +\] + +\end_inset + +Če +\begin_inset Formula $\forall\left(x,y,z\right)\in P:\nabla f\left(x,y,z\right)\not=0\Rightarrow P$ +\end_inset + + ploskev v +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + +, saj je po izreku o implicitni fji +\begin_inset Formula $P$ +\end_inset + + lokalno graf fje dveh spremenljivk. + Normala tangentne ravnine v +\begin_inset Formula $\left(x,y,z\right)\in P$ +\end_inset + + ima normalo +\begin_inset Formula $\nabla f\left(x,y,z\right)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{multicols} +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/šola/ars/ol2.odt b/šola/ars/ol2.odt Binary files differnew file mode 100644 index 0000000..32a2d7d --- /dev/null +++ b/šola/ars/ol2.odt diff --git a/šola/ds2/kolokvij2.lyx b/šola/ds2/kolokvij2.lyx new file mode 100644 index 0000000..0d2e149 --- /dev/null +++ b/šola/ds2/kolokvij2.lyx @@ -0,0 +1,3368 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\begin_preamble +\usepackage{siunitx} +\usepackage{pgfplots} +\usepackage{listings} +\usepackage{multicol} +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}} +\DeclareMathOperator{\g}{g} +\DeclareMathOperator{\sled}{sled} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Cir}{Cir} +\DeclareMathOperator{\ecc}{ecc} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\diam}{diam} +\newcommand\euler{e} +\end_preamble +\use_default_options true +\begin_modules +enumitem +\end_modules +\maintain_unincluded_children false +\language slovene +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics xetex +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification false +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 1cm +\topmargin 1cm +\rightmargin 1cm +\bottommargin 2cm +\headheight 1cm +\headsep 1cm +\footskip 1cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style german +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +setlength{ +\backslash +columnseprule}{0.2pt} +\backslash +begin{multicols}{2} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Največja stopnja +\begin_inset Formula $\Delta G$ +\end_inset + +, najmanjša +\begin_inset Formula $\delta G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Rokovanje: +\begin_inset Formula $\sum_{v\in VG}\deg_{G}v=2\left|EG\right|$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Vsak graf ima sodo mnogo vozlišč lihe stopnje. +\end_layout + +\begin_layout Standard +Presek/unija grafov je presek/unija njunih +\begin_inset Formula $V$ +\end_inset + + in +\begin_inset Formula $E$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $G\cup H$ +\end_inset + + je disjunktna unija grafov +\begin_inset Formula $\sim$ +\end_inset + + +\begin_inset Formula $VG\cap VH=\emptyset$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Komplement grafa: +\begin_inset Formula $\overline{G}$ +\end_inset + + (obratna povezanost) +\end_layout + +\begin_layout Standard +\begin_inset Formula $0\leq\left|EG\right|\leq{\left|VG\right| \choose 2}\quad\quad\quad$ +\end_inset + +Za padajoče +\begin_inset Formula $d_{i}$ +\end_inset + + velja: +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(d_{1},\dots d_{n}\right)$ +\end_inset + + graf +\begin_inset Formula $\Leftrightarrow\left(d_{2}-1,\dots,d_{d_{1}+1}-1,\dots,d_{n}\right)$ +\end_inset + + graf +\end_layout + +\begin_layout Standard +Če je +\begin_inset Formula $AG$ +\end_inset + + matrika sosednosti, +\begin_inset Formula $\left(\left(AG\right)^{n}\right)_{i,j}$ +\end_inset + + pove št. + +\begin_inset Formula $i,j-$ +\end_inset + +poti. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\text{Število trikotnikov: }\frac{\sled\left(\left(AG\right)^{3}\right)}{2\cdot3} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left|EK_{n}\right|={n \choose 2}$ +\end_inset + +, +\begin_inset Formula ${n \choose k}=\frac{n!}{k!\left(n-k\right)!}$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Sprehod +\end_layout + +\begin_layout Standard +je zaporedje vozlišč, ki so verižno povezana. +\end_layout + +\begin_layout Standard +Dolžina sprehoda je število prehojenih povezav. +\end_layout + +\begin_layout Standard +Sklenjen sprehod dolžine +\begin_inset Formula $k$ +\end_inset + +: +\begin_inset Formula $v_{0}=v_{k}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Enostaven sprehod ima disjunktna vozlišča razen +\begin_inset Formula $\left(v_{0},v_{k}\right)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Pot v grafu: podgraf +\begin_inset Formula $P_{k}$ +\end_inset + + +\begin_inset Formula $\sim$ +\end_inset + + enostaven nesklenjen sprehod. +\end_layout + +\begin_layout Paragraph +Cikel +\end_layout + +\begin_layout Standard +podgraf, ki je enostaven sklenjen sprehod dolžine +\begin_inset Formula $>3$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Če v +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $\exists$ +\end_inset + + dve različni +\begin_inset Formula $u,v-$ +\end_inset + +poti +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + premore cikel +\end_layout + +\begin_layout Standard +Sklenjen sprehod lihe dolžine +\begin_inset Formula $\in G$ +\end_inset + + +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + lih cikel +\begin_inset Formula $\in G$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Povezanost +\end_layout + +\begin_layout Standard +\begin_inset Formula $u,v$ +\end_inset + + sta v isti komponenti +\begin_inset Formula $\text{\ensuremath{\sim}}$ +\end_inset + + +\begin_inset Formula $\text{\ensuremath{\exists}}$ +\end_inset + + +\begin_inset Formula $u,v-$ +\end_inset + +pot +\end_layout + +\begin_layout Standard +Število komponent grafa: +\begin_inset Formula $\Omega G$ +\end_inset + +. + +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\sim\Omega G=1$ +\end_inset + + +\end_layout + +\begin_layout Standard +Komponenta je maksimalen povezan podgraf. +\end_layout + +\begin_layout Standard +Premer: +\begin_inset Formula $\diam G=\max\left\{ d_{G}\left(v,u\right);\forall v,u\in VG\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +Ekscentričnost: +\begin_inset Formula $\ecc_{G}u=max\left\{ d_{G}\left(u,x\right);\forall x\in VG\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +Polmer: +\begin_inset Formula $\rad G=\min\left\{ \ecc u;\forall u\in VG\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\diam C_{n}=\rad C_{n}=\lfloor\frac{n}{2}\rfloor$ +\end_inset + + +\end_layout + +\begin_layout Standard +(Liha) ožina ( +\begin_inset Formula $\g G$ +\end_inset + +) je dolžina najkrajšega (lihega) cikla. +\end_layout + +\begin_layout Standard +Vsaj en od +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $\overline{G}$ +\end_inset + + je povezan. +\end_layout + +\begin_layout Standard +Povezava +\begin_inset Formula $e$ +\end_inset + + je most +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + +\begin_inset Formula $\Omega\left(G-e\right)>\Omega G$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $u$ +\end_inset + + je prerezno vozlišče +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + +\begin_inset Formula $\Omega\left(G-u\right)>\Omega G$ +\end_inset + + +\end_layout + +\begin_layout Standard +Za nepovezan +\begin_inset Formula $G$ +\end_inset + + velja +\begin_inset Formula $\diam\overline{G}\leq2$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Dvodelni +\end_layout + +\begin_layout Standard +\begin_inset Formula $\sim V=A\cup B,A\cap B=\emptyset,\forall uv\in E:u\in A\oplus v\in A$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $K_{m,n}$ +\end_inset + + je poln dvodelni graf, +\begin_inset Formula $\left|A\right|=m$ +\end_inset + +, +\begin_inset Formula $\left|B\right|=n$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + dvodelen +\begin_inset Formula $\Leftrightarrow\forall$ +\end_inset + + komponenta +\begin_inset Formula $G$ +\end_inset + + dvodelna +\end_layout + +\begin_layout Standard +Pot, sod cikel, hiperkocka so dvodelni, petersenov ni. +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + dvodelen +\begin_inset Formula $\Leftrightarrow G$ +\end_inset + + ne vsebuje lihega cikla. +\end_layout + +\begin_layout Standard +Biparticija glede na parnost +\begin_inset Formula $d_{G}\left(u,x_{0}\right)$ +\end_inset + +, +\begin_inset Formula $x_{0}$ +\end_inset + + fiksen. +\end_layout + +\begin_layout Standard +Dvodelen +\begin_inset Formula $k-$ +\end_inset + +regularen, +\begin_inset Formula $\left|E\right|\ge1\Rightarrow$ +\end_inset + + +\begin_inset Formula $\left|A\right|=\left|B\right|$ +\end_inset + +. + Dokaz: +\begin_inset Formula $\sum_{u\in A}\deg u=\left|E\right|=\cancel{k}\left|A\right|=\cancel{k}\left|B\right|=\sum_{u\in B}\deg u$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Krožni +\end_layout + +\begin_layout Standard +\begin_inset Formula $\Cir\left(n,\left\{ s_{1},\dots,s_{k}\right\} \right):\left|V\right|=n,$ +\end_inset + + množica preskokov +\end_layout + +\begin_layout Standard +\begin_inset Formula $\Omega\Cir\left(n,\left\{ s,n-s\right\} \right)=\gcd\left\{ n,s\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $W_{n}$ +\end_inset + + (kolo) pa je cikel z univerzalnim vozliščem. +\end_layout + +\begin_layout Paragraph +Homomorfizem +\end_layout + +\begin_layout Standard +\begin_inset Formula $\varphi:VG\to VH$ +\end_inset + + slika povezave v povezave +\end_layout + +\begin_layout Standard +Primer: +\begin_inset Formula $K_{2}$ +\end_inset + + je homomorfna slika +\begin_inset Formula $\forall$ +\end_inset + + bipartitnega grafa. +\end_layout + +\begin_layout Standard +V povezavah in vozliščih surjektiven +\begin_inset Formula $hm\varphi$ +\end_inset + + je epimorfizem. +\end_layout + +\begin_layout Standard +V vozliščih injektiven +\begin_inset Formula $hm\varphi$ +\end_inset + + je monomorfizem/vložitev. +\end_layout + +\begin_layout Standard +Vložitev, ki ohranja razdalje, je izometrična. +\end_layout + +\begin_layout Standard +Kompozitum homomorfizmov je spet homomorfizem. +\end_layout + +\begin_layout Standard +Izomorfizem je bijektivni +\begin_inset Formula $hm\varphi$ +\end_inset + + s homomorfnim inverzom. +\end_layout + +\begin_layout Standard +\begin_inset Formula $im\varphi$ +\end_inset + + +\begin_inset Formula $f:VG\to VH$ +\end_inset + + +\begin_inset Formula $\forall u,v\in VG:uv\in EG\Leftrightarrow fufv\in EH$ +\end_inset + + +\end_layout + +\begin_layout Standard +Nad množico vseh grafov +\begin_inset Formula $\mathcal{G}$ +\end_inset + + je izomorfizem ( +\begin_inset Formula $\cong$ +\end_inset + +) ekv. + rel. +\end_layout + +\begin_layout Standard +\begin_inset Formula $im\varphi$ +\end_inset + + +\begin_inset Formula $G\to G$ +\end_inset + + je avtomorfizem. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\Aut G$ +\end_inset + + je grupa avtomorfizmov s komponiranjem. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\Aut K_{n}=\left\{ \pi\in S_{n}=\text{permutacije }n\text{ elementov}\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\Aut P_{n}=\left\{ \text{trivialni }id,f\left(i\right)=n-i-1\right\} $ +\end_inset + +, +\begin_inset Formula $\Aut G\cong\Aut\overline{G}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Izomorfizem ohranja stopnje, št. + +\begin_inset Formula $C_{4}$ +\end_inset + +, povezanost, +\begin_inset Formula $\left|EG\right|,\dots$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G\cong\overline{G}\Rightarrow\left|VG\right|\%4\in\left\{ 0,1\right\} $ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Operacije +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + vpeti podgraf +\begin_inset Formula $G\Leftrightarrow\exists F\subseteq EG\ni:H=G-F$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + inducirani podgraf +\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG\ni:H=G-S$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + podgraf +\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG,F\subseteq EG\ni:H=\left(G-F\right)-S$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $uv=e\in EG$ +\end_inset + +. + +\begin_inset Formula $G/e$ +\end_inset + + je skrčitev. + (identificiramo +\begin_inset Formula $u=v$ +\end_inset + +) +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + minor +\begin_inset Formula $G$ +\end_inset + +: +\begin_inset Formula $H=f_{1}...f_{k}G$ +\end_inset + + za +\begin_inset Formula $f_{i}$ +\end_inset + + skrčitev/odstranjevanje +\end_layout + +\begin_layout Standard +\begin_inset Formula $VG^{+}e\coloneqq VG\cup\left\{ x_{e}\right\} $ +\end_inset + +, +\begin_inset Formula $EG^{+}e\coloneqq EG\setminus e\cup\left\{ x_{e}u,x_{e}v\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $VG^{+}e$ +\end_inset + + je subdivizija, +\begin_inset Formula $e\in EG$ +\end_inset + +. + Na +\begin_inset Formula $e$ +\end_inset + + dodamo vozlišče. +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + subdivizija +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $\Leftrightarrow H=G^{+}\left\{ e_{1},\dots,e_{k}\right\} ^{+}\left\{ f_{1}\dots f_{j}\right\} ^{+}\dots$ +\end_inset + + +\end_layout + +\begin_layout Standard +Stopnja vozlišč se s subdivizijo ne poveča. +\end_layout + +\begin_layout Standard +Glajenje +\begin_inset Formula $G^{-}v$ +\end_inset + +, +\begin_inset Formula $v\in VG$ +\end_inset + + je obrat subdivizije. + +\begin_inset Formula $\deg_{G}v=2$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $H$ +\end_inset + + sta homeomorfna, če sta subdivizija istega grafa. +\end_layout + +\begin_layout Standard +Kartezični produkt: +\begin_inset Formula $V\left(G\square H\right)\coloneqq VG\times VH$ +\end_inset + +, +\begin_inset Formula $E\left(G\square H\right)\coloneqq\left\{ \left\{ \left(g,h\right),\left(g',h'\right)\right\} ;g=g'\wedge hh'\in EH\vee h=h'\wedge gg'\in EG\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(\mathcal{G},\square\right)$ +\end_inset + + monoid, enota +\begin_inset Formula $K_{1}$ +\end_inset + +. + +\begin_inset Formula $Q_{n}\cong Q_{n-1}\square K_{2}=Q_{n-2}\square K_{2}^{\square,2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Disjunktna unija: +\begin_inset Formula $G$ +\end_inset + +, +\begin_inset Formula $H$ +\end_inset + + disjunktna. + +\begin_inset Formula $V\left(G\cup H\right)\coloneqq VG\cup VH$ +\end_inset + +, +\begin_inset Formula $E\left(G\cup H\right)\coloneqq EG\cup EH$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G,H$ +\end_inset + + dvodelna +\begin_inset Formula $\Rightarrow G\square H$ +\end_inset + + dvodelen +\end_layout + +\begin_layout Paragraph +\begin_inset Formula $k-$ +\end_inset + +povezan graf +\end_layout + +\begin_layout Standard +ima +\begin_inset Formula $\geq k+1$ +\end_inset + + vozlišč in ne vsebuje prerezne množice moči +\begin_inset Formula $<k$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $X\subseteq VG$ +\end_inset + + je prerezna množica +\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $Y\subseteq EG$ +\end_inset + + prerezna množica povezav +\begin_inset Formula $\Leftrightarrow\Omega\left(G-Y\right)>\Omega G$ +\end_inset + + +\end_layout + +\begin_layout Standard +Povezanost grafa: +\begin_inset Formula $\kappa G=\max k$ +\end_inset + +, da je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +povezan. + Primeri: +\begin_inset Formula $\kappa K_{n}=n-1$ +\end_inset + +, +\begin_inset Formula $\kappa P_{n}=1$ +\end_inset + +, +\begin_inset Formula $\kappa C_{n}=2$ +\end_inset + +, +\begin_inset Formula $\kappa K_{n,m}=\min\left\{ n,m\right\} $ +\end_inset + +, +\begin_inset Formula $\kappa Q_{n}=n$ +\end_inset + +, +\begin_inset Formula $\kappa G\text{\ensuremath{\leq}}\delta G$ +\end_inset + + +\end_layout + +\begin_layout Standard +Izrek (Menger): +\begin_inset Formula $\left|VG\right|\geq k+1$ +\end_inset + +: +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +povezan +\begin_inset Formula $\Leftrightarrow\forall u,v\in VG,uv\not\in EG:\exists k$ +\end_inset + + notranje disjunktnih +\begin_inset Formula $u,v-$ +\end_inset + +poti +\end_layout + +\begin_layout Standard +Graf je +\begin_inset Formula $k-$ +\end_inset + +povezan po povezavah, če ne vsebuje prerezne množice povezav moči +\begin_inset Formula $<k$ +\end_inset + +. + Povezanost grafa po povezavah: +\begin_inset Formula $\kappa'G=\max k$ +\end_inset + +, da je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +povezan po povezavah. + Primeri: +\begin_inset Formula $\kappa'K_{n}=n-1$ +\end_inset + +, +\begin_inset Formula $\kappa'P_{n}=1$ +\end_inset + +, +\begin_inset Formula $\kappa'C_{n}=2$ +\end_inset + + +\end_layout + +\begin_layout Standard +Izrek (Menger'): +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +povezan +\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$ +\end_inset + + po povezavah disjunktnih +\begin_inset Formula $u,v-$ +\end_inset + +poti +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall G\in\mathcal{G}:\kappa G\leq\kappa'G\leq\delta G$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Drevo +\end_layout + +\begin_layout Standard +je povezan gozd. + Gozd je graf brez ciklov. +\end_layout + +\begin_layout Standard +Drevo z vsaj dvema vozliščema premore dva lista. +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +hspace*{ +\backslash +fill} +\end_layout + +\end_inset + +NTSE: +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + drevo +\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists!u,v-$ +\end_inset + +pot +\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\forall e\in EG:e$ +\end_inset + + most +\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\left|EG\right|=\left|VG\right|-1$ +\end_inset + + +\end_layout + +\begin_layout Standard +Vpeto drevo grafa je vpet podgraf, ki je drevo. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\tau G\sim$ +\end_inset + + število vpetih dreves. + +\begin_inset Formula $\Omega G=1\Leftrightarrow\tau G\geq1$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall e\in EG:\tau G=\tau\left(G-e\right)+\tau\left(G/e\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\text{\text{\text{Laplaceova matrika: }}}L\left(G\right)_{ij}=\begin{cases} +\deg_{G}v_{i} & ;i=j\\ +-\left(\text{št. uv povezav}\right) & ;\text{drugače} +\end{cases} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Izrek (Kirchoff): za +\begin_inset Formula $G$ +\end_inset + + povezan multigraf je +\begin_inset Formula $\forall i:\tau G=\det\left(LG\text{ brez \ensuremath{i}te vrstice in \ensuremath{i}tega stolpca}\right)$ +\end_inset + +. + +\begin_inset Formula $\tau K_{n}=n^{n-2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Prüferjeva koda, če lahko linearno uredimo vozlišča: Ponavljaj, dokler +\begin_inset Formula $VG\not=\emptyset$ +\end_inset + +: vzemi prvi list, ga odstrani in v vektor dodaj njegovega soseda. +\end_layout + +\begin_layout Standard +Blok je maksimalen povezan podgraf brez prereznih vozlišč. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\tau G=\tau B_{1}\cdot\cdots\cdot\tau B_{k}$ +\end_inset + + za bloke +\begin_inset Formula $\vec{B}$ +\end_inset + + grafa +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{multicols} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{multicols}{2} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Paragraph +Eulerjev +\end_layout + +\begin_layout Standard +sprehod v m.grafu vsebuje vse povezave po enkrat. +\end_layout + +\begin_layout Standard +Eulerjev obhod je sklenjen eulerjev sprehod. +\end_layout + +\begin_layout Standard +Eulerjev graf premore eulerjev obhod. +\end_layout + +\begin_layout Standard +Za povezan multigraf +\begin_inset Formula $G$ +\end_inset + + eulerjev +\begin_inset Formula $\Leftrightarrow\forall v\in VG:\deg_{G}v$ +\end_inset + + sod +\end_layout + +\begin_layout Standard +Fleuryjev algoritem za eulerjev obhod v eulerjevem grafu: Začnemo v poljubni + povezavi, jo izbrišemo, nadaljujemo na mostu le, če nimamo druge možne + povezave. +\end_layout + +\begin_layout Standard +Dekompozicija: delitev na povezavno disjunktne podgrafe. +\end_layout + +\begin_layout Standard +Dekompozicija je lepa, če so podgrafi izomorfni. + ( +\begin_inset Formula $\exists$ +\end_inset + + za +\begin_inset Formula $P_{5,2}$ +\end_inset + +) +\end_layout + +\begin_layout Standard +Vsak eulerjev graf premore dekompozicijo v cikle. +\end_layout + +\begin_layout Standard +\begin_inset Formula $Q_{n}$ +\end_inset + + eulerjev +\begin_inset Formula $\Leftrightarrow n$ +\end_inset + + sod +\end_layout + +\begin_layout Standard +\begin_inset Formula $K_{m,n,p}$ +\end_inset + + eulerjev +\begin_inset Formula $\Leftrightarrow m,n,p$ +\end_inset + + iste parnosti +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + eulerjev +\begin_inset Formula $\wedge H$ +\end_inset + + eulerjev +\begin_inset Formula $\Rightarrow G\square H$ +\end_inset + + eulerjev +\end_layout + +\begin_layout Paragraph +Hamiltonov +\end_layout + +\begin_layout Standard +cikel vsebuje vsa vozlišča grafa. +\end_layout + +\begin_layout Standard +Hamilton graf premore Hamiltonov cikel. +\end_layout + +\begin_layout Standard +Hamiltonova pot vsebuje vsa vozlišča. +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + hamiltonov, +\begin_inset Formula $S\subseteq VG\Rightarrow\Omega\left(G-S\right)\leq\left|S\right|$ +\end_inset + + torej: +\end_layout + +\begin_layout Standard +\begin_inset Formula $\exists S\in VG:\Omega\left(G-S\right)>\left|S\right|\Rightarrow G$ +\end_inset + + ni hamiltonov. + Primer: +\begin_inset Formula $G$ +\end_inset + + vsebuje prerezno vozlišče +\begin_inset Formula $\Rightarrow G$ +\end_inset + + ni hamiltonov. +\end_layout + +\begin_layout Standard +\begin_inset Formula $K_{n,m}$ +\end_inset + + je hamiltonov +\begin_inset Formula $\Leftrightarrow m=n$ +\end_inset + + (za +\begin_inset Formula $S$ +\end_inset + + vzamemo +\begin_inset Formula $\min\left\{ m,n\right\} $ +\end_inset + +) +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left|VG\right|\geq3,\forall u,v\in VG:\deg_{G}u+\deg_{G}v\geq\left|VG\right|\Rightarrow G$ +\end_inset + + hamil. +\end_layout + +\begin_layout Standard +Dirac: +\begin_inset Formula $\left|VG\right|\geq3,\forall u\in VG:\deg_{G}u\geq\frac{\left|VG\right|}{2}\Rightarrow G$ +\end_inset + + hamilton. +\end_layout + +\begin_layout Paragraph +Ravninski +\end_layout + +\begin_layout Standard +graf brez sekanja povezav narišemo v ravnino +\end_layout + +\begin_layout Standard +\begin_inset Formula $K_{2,3}$ +\end_inset + + je ravninski, +\begin_inset Formula $K_{3,3}$ +\end_inset + +, +\begin_inset Formula $K_{5}$ +\end_inset + +, +\begin_inset Formula $C_{5}\square C_{5}$ +\end_inset + + in +\begin_inset Formula $P_{5,2}$ +\end_inset + + niso ravninski. +\end_layout + +\begin_layout Standard +Vložitev je ravninski graf z ustrezno risbo v ravnini. +\end_layout + +\begin_layout Standard +Lica so sklenjena območja vložitve brez vozlišč in povezav. +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + lahko vložimo v ravnino +\begin_inset Formula $\Leftrightarrow G$ +\end_inset + + lahko vložimo na sfero. +\end_layout + +\begin_layout Standard +Dolžina lica +\begin_inset Formula $F\sim lF$ +\end_inset + + je št. + povezav obhoda lica. +\end_layout + +\begin_layout Standard +Drevo je ravninski graf z enim licem dolžine +\begin_inset Formula $2\left|ET\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\sum_{F\in FG}lF=2\left|EG\right|$ +\end_inset + +, +\begin_inset Formula $lF\geq gG$ +\end_inset + + za ravninski +\begin_inset Formula $G$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $2\left|EG\right|=\sum_{F\in FG}lF\geq\sum_{F\in FG}gG=gG\left|FG\right|$ +\end_inset + + ( +\begin_inset Formula $G$ +\end_inset + + ravn.) +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + ravninski +\begin_inset Formula $\Rightarrow\left|EG\right|\geq\frac{gG}{2}$ +\end_inset + + +\begin_inset Formula $\left|FG\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +Euler: +\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=1+\Omega G$ +\end_inset + + za ravninski +\begin_inset Formula $G$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + ravninski, +\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\leq3\left|VG\right|-6$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + ravninski brez +\begin_inset Formula $C_{3}$ +\end_inset + +, +\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\text{\ensuremath{\leq2\left|VG\right|-4}}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Triangulacija je taka vložitev, da so vsa lica omejena s +\begin_inset Formula $C_{3}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +V maksimalen ravninski graf ne moremo dodati povezave, da bi ostal ravninski. + +\begin_inset Formula $\sim$ +\end_inset + + Ni pravi vpet podgraf ravn. + grafa. +\end_layout + +\begin_layout Standard +\begin_inset Formula $K_{5}-e$ +\end_inset + + je maksimalen ravninski graf +\begin_inset Formula $\forall e\in EK_{5}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + ravninski. + NTSE: +\begin_inset Formula $G$ +\end_inset + + triangulacija +\begin_inset Formula $\Leftrightarrow G$ +\end_inset + + maksimalen ravninski +\begin_inset Formula $\Leftrightarrow\left|EG\right|=3\left|VG\right|-6$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + ravninski +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + vsaka subdivizija +\begin_inset Formula $G$ +\end_inset + + ravninska. +\end_layout + +\begin_layout Standard +Kuratovski: +\begin_inset Formula $G$ +\end_inset + + ravn. + +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + ne vsebuje subdivizije +\begin_inset Formula $K_{5}$ +\end_inset + + ali +\begin_inset Formula $K_{3,3}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Wagner: +\begin_inset Formula $G$ +\end_inset + + ravn. + +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + niti +\begin_inset Formula $K_{5}$ +\end_inset + + niti +\begin_inset Formula $K_{3,3}$ +\end_inset + + nista njegova minorja +\end_layout + +\begin_layout Standard +Zunanje-ravninski ima vsa vozlišča na robu zunanjega lica. +\end_layout + +\begin_layout Standard +\begin_inset Formula $2-$ +\end_inset + +povezan zunanje-ravninski je +\series bold +hamiltonov +\series default +. +\end_layout + +\begin_layout Standard +Zunanje-ravninski +\begin_inset Formula $G$ +\end_inset + +, +\begin_inset Formula $\left|VG\right|\geq2$ +\end_inset + + ima vozlišče stopnje +\begin_inset Formula $\leq2$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph +Barvanje vozlišč +\end_layout + +\begin_layout Standard +je taka +\begin_inset Formula $C:VG\to\left\{ 1..k\right\} \Leftrightarrow\forall uv\in EG:Cu\not=Cv$ +\end_inset + + +\end_layout + +\begin_layout Standard +Kromatično število +\begin_inset Formula $\chi G$ +\end_inset + + je najmanjši +\begin_inset Formula $k$ +\end_inset + +, +\begin_inset Formula $\ni:\exists$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +barvanje +\begin_inset Formula $G$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\chi C_{n}=\begin{cases} +2 & ;n\text{ sod}\\ +3 & ;n\text{ lih} +\end{cases}$ +\end_inset + +, +\begin_inset Formula $\chi G\leq\left|VG\right|,\chi G=\left|VG\right|\Leftrightarrow G\cong K_{n}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + podgraf +\begin_inset Formula $G\Rightarrow\chi G\geq\chi H$ +\end_inset + +. + +\begin_inset Formula $\chi\text{bipartitni}=2$ +\end_inset + + +\end_layout + +\begin_layout Standard +Barvni razred so vsa vozlišča iste barve. + Je brez povezav +\begin_inset Formula $\sim$ +\end_inset + + +\series bold +neodvisna množica +\series default +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\chi G=k\Leftrightarrow\chi G\leq k\wedge\chi G\geq k$ +\end_inset + + +\end_layout + +\begin_layout Standard +Klično št.: +\begin_inset Formula $\omega G\coloneqq$ +\end_inset + + +\begin_inset Formula $\left|V\right|$ +\end_inset + + najv. + polnega podgr. + v +\begin_inset Formula $G$ +\end_inset + +. + +\begin_inset Formula $\omega G\leq\chi G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Trditev: +\begin_inset Formula $\forall n\in\mathbb{N}\exists G\in\mathcal{G}\ni:\omega G=2\wedge\chi G=n$ +\end_inset + + +\end_layout + +\begin_layout Standard +Požrešno barvanje: Vozlišča v poljubnem vrstnem redu barvamo z najnižno + možno barvo. +\end_layout + +\begin_layout Standard +Vedno +\begin_inset Formula $\exists$ +\end_inset + + vrstni red, ki vrne barvanje s +\begin_inset Formula $\chi G$ +\end_inset + + barvami. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall G\in\mathcal{G}:\chi G\leq\Delta G+1$ +\end_inset + +, če +\begin_inset Formula $G$ +\end_inset + + ni regularen celo +\begin_inset Formula $\chi G\leq\Delta G$ +\end_inset + + +\end_layout + +\begin_layout Standard +Brooks: +\begin_inset Formula $G$ +\end_inset + + povezan, +\begin_inset Formula $G$ +\end_inset + + ni poln niti lihi cikel +\begin_inset Formula $\Rightarrow\chi G\leq\Delta G$ +\end_inset + + +\end_layout + +\begin_layout Standard +let +\begin_inset Formula $d_{1}\geq\cdots\geq d_{n}$ +\end_inset + + stopnje. + +\begin_inset Formula $\chi G=1+\max_{i=1}^{n}\min\left\{ d_{i},i-1\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +Spoj +\begin_inset Formula $G\oplus H$ +\end_inset + + je disjunktna unija +\begin_inset Formula $G$ +\end_inset + +, +\begin_inset Formula $H$ +\end_inset + + z vsemi pov. + med njima +\end_layout + +\begin_layout Standard +Disjunktna unija je unija brez preseka. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\chi\left(G\oplus H\right)=\chi G+\chi H$ +\end_inset + + +\end_layout + +\begin_layout Standard +Ravninski graf ima +\begin_inset Formula $\chi G\leq4$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Barvanje povezav +\end_layout + +\begin_layout Standard +Povezavi s skupnim vozl. + +\begin_inset Formula $\Rightarrow$ +\end_inset + + razl. + barvi. +\end_layout + +\begin_layout Standard +Kromatični indeks +\begin_inset Formula $\chi'G$ +\end_inset + + je najm. + št. + barv za barv. + pov. + +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\chi'C_{n}=\begin{cases} +2 & ;n\text{ sod}\\ +3 & ;n\text{ lih} +\end{cases}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Vizing: +\begin_inset Formula $\forall G\in\mathcal{G}:\Delta G\geq\chi'G\geq\Delta G+1$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\chi G\in\left\{ \Delta G\text{ razred I.},\Delta G+1\text{ razred II.}\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $K_{2k+1}\in$ +\end_inset + + II., +\begin_inset Formula $K_{2k}\in$ +\end_inset + + I., bipartitni +\begin_inset Formula $\in$ +\end_inset + + I. +\end_layout + +\begin_layout Paragraph +Neodvisne množice +\end_layout + +\begin_layout Standard +\begin_inset Formula $I\subseteq VG$ +\end_inset + + je neodvisna, če je podgraf, induciran z +\begin_inset Formula $I$ +\end_inset + +, brez povezav. +\end_layout + +\begin_layout Standard +Neodvisnostno št +\begin_inset Formula $\alpha G$ +\end_inset + + je moč največje neodvisne množice. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\alpha K_{n}=1$ +\end_inset + +, +\begin_inset Formula $\alpha C_{n}=\lfloor\frac{n}{2}\rfloor$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\alpha G\leq\sum_{i=1}^{k}\alpha H_{i}$ +\end_inset + +, kjer so +\begin_inset Formula $H_{i}$ +\end_inset + + disjunktni inducirani podgr. + +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall G\in\mathcal{G}:\alpha G\cdot\chi G\geq\left|VG\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\frac{\left|VG\right|}{\chi G}\leq\alpha G\leq\left|VG\right|-\frac{\left|EG\right|}{\Delta G}$ +\end_inset + + — zgornja in spodnja meja. +\end_layout + +\begin_layout Standard +\begin_inset Formula $Iu$ +\end_inset + + je +\begin_inset Formula $\alpha$ +\end_inset + + poddrevesa s korenom +\begin_inset Formula $u$ +\end_inset + + v +\begin_inset Formula $T$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\alpha T=Ir=\max\left\{ 1+\sum_{u\text{ sinovi }r}Iu,\sum_{u\text{ vnuki }r}Iu\right\} $ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Dominantne množice +\end_layout + +\begin_layout Standard +Neodvisna +\begin_inset Formula $I\subseteq G$ +\end_inset + + je maksimalna +\begin_inset Formula $\sim\nexists S\text{ neodvisna}\subseteq G\ni:I\subset S\sim$ +\end_inset + + ni prava +\begin_inset Formula $\subset$ +\end_inset + + neodv. + mn. +\end_layout + +\begin_layout Standard +Dominantna množica je maksimalna neodvisna, kjer ima vsako vozlišče iz +\begin_inset Formula $G\setminus I$ +\end_inset + + soseda v +\begin_inset Formula $I$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $D\subseteq VG$ +\end_inset + + dominira +\begin_inset Formula $X\subseteq VG$ +\end_inset + +, če je vsako vozlišče iz +\begin_inset Formula $X$ +\end_inset + + v +\begin_inset Formula $D$ +\end_inset + + ali pa ima soseda v +\begin_inset Formula $D$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $N_{G}\left(u\right)=$ +\end_inset + +sosedje +\begin_inset Formula $u$ +\end_inset + + v +\begin_inset Formula $G$ +\end_inset + +, +\begin_inset Formula $N_{G}\left[u\right]=N_{G}\left(u\right)\cup\left\{ u\right\} $ +\end_inset + + zap. + sos. + +\begin_inset Formula $u$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $N_{G}\left[D\right]=\bigcup_{u\in D}N_{G}\left[u\right]$ +\end_inset + + zaprta soseščina množice +\begin_inset Formula $D$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $D$ +\end_inset + + dominira +\begin_inset Formula $X\sim X\subseteq N_{G}\left[D\right]$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $D$ +\end_inset + + dominira +\begin_inset Formula $VG\sim D$ +\end_inset + + dominantna množica +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Dominacijsko število +\begin_inset Formula $\gamma G=$ +\end_inset + + moč največje dom. + mn. +\end_layout + +\begin_layout Standard +Vsaka maksimalna neodvisna mn. + +\begin_inset Formula $G$ +\end_inset + + je dom. + mn. + +\begin_inset Formula $G$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\gamma K_{n}=1$ +\end_inset + +, +\begin_inset Formula $\gamma C_{n}=\lceil\frac{n}{3}\rceil$ +\end_inset + +, +\begin_inset Formula $\gamma P=3$ +\end_inset + + +\end_layout + +\begin_layout Standard +Za vsak graf brez izoliranih vozlišč +\begin_inset Formula $\lceil\frac{\left|VG\right|}{\Delta G+1}\rceil\leq\gamma G\leq\lfloor\frac{\left|VG\right|}{2}\rfloor$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $X$ +\end_inset + + je 2-pakiranje +\begin_inset Formula $G$ +\end_inset + +, če +\begin_inset Formula $\forall x,y\in X:x\not=y\Rightarrow d_{G}\left(x,y\right)\geq3$ +\end_inset + + zdb +\begin_inset Formula $\forall x,y\in X:x\not=y\Rightarrow N_{G}x\cap N_{G}y=\emptyset$ +\end_inset + + +\end_layout + +\begin_layout Standard +Moč največjega 2-pakiranja je 2-pakirno število +\begin_inset Formula $\rho G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\Rightarrow\gamma G\geq\rho G$ +\end_inset + + +\end_layout + +\begin_layout Standard +Dominantna +\begin_inset Formula $X$ +\end_inset + + je popolna koda +\begin_inset Formula $G\Leftrightarrow\bigcup_{u\in X}N_{G}\left[u\right]=VG$ +\end_inset + + +\end_layout + +\begin_layout Standard +Če graf premore popolno kodo +\begin_inset Formula $\Rightarrow\gamma G=\rho G$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + vpet podgraf +\begin_inset Formula $G\Rightarrow\gamma G\leq\gamma H$ +\end_inset + + +\end_layout + +\begin_layout Standard +Povezan +\begin_inset Formula $G$ +\end_inset + + premore vpeto drevo +\begin_inset Formula $T\ni:\gamma G=\gamma T$ +\end_inset + +. + Dokaz: odstrani vse povezave, kjer +\begin_inset Formula $G-e$ +\end_inset + + ohrani +\begin_inset Formula $\gamma$ +\end_inset + + in povezanost. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left|VG\right|\geq1\Rightarrow\gamma G\leq\chi\overline{G}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Povezana dom. + mn. + inducira povez. + podgraf. + — +\begin_inset Formula $\gamma_{c}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Neodvisna dom. + mn. + inducira podgraf brez povezav — +\begin_inset Formula $\gamma_{i}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $X$ +\end_inset + + je celotna dom. + mn. + +\begin_inset Formula $\Leftrightarrow\forall v\in VG:N_{G}\left(v\right)\cap X\not=\emptyset$ +\end_inset + + — +\begin_inset Formula $\gamma_{t}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $G$ +\end_inset + + brez izol. + vozl.: +\begin_inset Formula $\gamma G\leq\gamma_{t}G\leq2\gamma G$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Algebrske strukture z eno operacijo +\end_layout + +\begin_layout Standard +Grupoid: notranjost, polgrupa: asociativen grupoid, monoid: polgrupa z enoto +\end_layout + +\begin_layout Standard +Grupa: +\begin_inset Formula $\forall a\in A\exists a^{-1}\in A\ni:a\cdot a^{-1}=a^{-1}\cdot a=e$ +\end_inset + + +\end_layout + +\begin_layout Standard +Potence v monoidu: +\begin_inset Formula $n\in\mathbb{N}_{0}$ +\end_inset + +. + +\begin_inset Formula $a^{0}=e$ +\end_inset + +, +\begin_inset Formula $a^{n}=a\cdot a^{n-1}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(a\cdot b\right)^{-1}=b^{-1}\cdot a^{-1}$ +\end_inset + +, +\begin_inset Formula $a^{n}\cdot a^{m}=a^{n+m},\left(a^{n}\right)^{m}=a^{nm}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(a^{-1}\right)^{n}=\left(a^{n}\right)^{-1}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(\mathbb{Z}_{m},+_{m}\right)$ +\end_inset + + grupa, +\begin_inset Formula $\left(\mathbb{Z}_{m},\cdot_{n}\right)$ +\end_inset + + monoid +\end_layout + +\begin_layout Standard +\begin_inset Formula $k\in\mathbb{Z}_{n}$ +\end_inset + + obrnljiv v +\begin_inset Formula $\left(\mathbb{Z}_{n},\cdot_{n}\right)\Leftrightarrow k\perp n\sim\gcd\left\{ k,n\right\} =1$ +\end_inset + + +\end_layout + +\begin_layout Standard +Komutativni grupi pravimo abelova. +\end_layout + +\begin_layout Standard +Cayleyeva tabela je predpis za grupoid +\begin_inset Formula $\cdot:A^{2}\to A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +red +\begin_inset Formula $a$ +\end_inset + + je najmanjši +\begin_inset Formula $n\in\mathbb{N}\ni:a^{n}=e$ +\end_inset + + oz. + +\begin_inset Formula $\infty$ +\end_inset + +, če ne obstaja. +\end_layout + +\begin_layout Standard +Def.: +\begin_inset Formula $H\subseteq G$ +\end_inset + + podgrupa, če je grupa za isto operacijo. +\end_layout + +\begin_layout Standard +Izrek: +\begin_inset Formula $H\subseteq G$ +\end_inset + + podgrupa +\begin_inset Formula $\Leftrightarrow\forall x,y\in H:x^{-1}y\in H$ +\end_inset + + +\end_layout + +\begin_layout Standard +Trivialni podgrupi +\begin_inset Formula $G$ +\end_inset + + sta +\begin_inset Formula $\left\{ e\right\} $ +\end_inset + + in +\begin_inset Formula $G$ +\end_inset + +. + +\begin_inset space \hfill{} +\end_inset + +Ciklična podgrupa: +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left\langle a\right\rangle \coloneqq\left\{ a^{n};\forall n\in\mathbb{Z}\right\} $ +\end_inset + + je podgrupa v +\begin_inset Formula $G$ +\end_inset + + za +\begin_inset Formula $a\in\text{grupa }G$ +\end_inset + + +\end_layout + +\begin_layout Standard +Center grupe: +\begin_inset Formula $ZG=\left\{ a\in G;\forall x\in G:ax=xa\right\} $ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(ZG,\cdot\right)$ +\end_inset + +je podgrupa grupe +\begin_inset Formula $\left(G,\cdot\right)$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph +Permutacijske grupe +\end_layout + +\begin_layout Standard +Permutacija je bijekcija +\begin_inset Formula $A\to A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +perm. + gr. + so permutacije +\begin_inset Formula $A$ +\end_inset + +, ki tvorijo grupo za +\begin_inset Formula $\circ$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Polna simetrična grupa +\begin_inset Formula $S_{n}\coloneqq\left\{ \pi:\left\{ 1..n\right\} \to\left\{ 1..n\right\} ;\pi\text{ bijekcija}\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +Alternirajoča grupa +\begin_inset Formula $A_{n}\coloneqq\left\{ \pi\in S_{n};\pi\text{ soda}\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +Permutacija kot produkt disjunktnih ciklov dolžin +\begin_inset Formula $l_{1},\dots,l_{n}$ +\end_inset + + je soda, če je +\begin_inset Formula $\left(l_{1}-1\right)+\cdots+\left(l_{n}-1\right)$ +\end_inset + + sod. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left|S_{n}\right|=n!$ +\end_inset + +, +\begin_inset Formula $\left|A_{n}\right|=\frac{n!}{2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(G,\circ\right)$ +\end_inset + +, +\begin_inset Formula $\left(H,*\right)$ +\end_inset + + grupi. + +\begin_inset Formula $f:G\to H$ +\end_inset + + je hm +\begin_inset Formula $\varphi\Leftrightarrow\forall x,y\in G:f\left(x\circ y\right)=fx*fy$ +\end_inset + +. + +\begin_inset Formula $f$ +\end_inset + + je še celo im +\begin_inset Formula $\varphi\Leftrightarrow f$ +\end_inset + + bijekcija. +\end_layout + +\begin_layout Standard +Grupi sta izomorfni +\begin_inset Formula $\sim\text{\ensuremath{\exists}}$ +\end_inset + +im +\begin_inset Formula $\varphi$ +\end_inset + + med njima: +\begin_inset Formula $G\approx H$ +\end_inset + + +\end_layout + +\begin_layout Standard +Vsaka grupa je izomorfna neki permutacijski grupi +\end_layout + +\begin_layout Paragraph +Odseki in podgrupe edinke +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(G,\circ\right)$ +\end_inset + + grupa, +\begin_inset Formula $H\subseteq G$ +\end_inset + +. + za +\begin_inset Formula $a\in G$ +\end_inset + +: +\begin_inset Formula $aH=\left\{ ah;h\in H\right\} ,Ha=\left\{ ha;h\in H\right\} $ +\end_inset + +. + Za +\begin_inset Formula $H$ +\end_inset + + podgrupo pravimo +\begin_inset Formula $aH$ +\end_inset + + levi odsek +\begin_inset Formula $H$ +\end_inset + + in +\begin_inset Formula $Ha$ +\end_inset + + desni. + Velja: +\begin_inset Formula $a\in aH$ +\end_inset + +, +\begin_inset Formula $aH=H\Leftrightarrow a\in H$ +\end_inset + +, +\begin_inset Formula $aH=bH\nabla aH\cap bH=\emptyset$ +\end_inset + +, +\begin_inset Formula $aH=bH\Leftrightarrow a^{-1}b\in H$ +\end_inset + +, +\begin_inset Formula $\left|aH\right|=\left|bH\right|$ +\end_inset + +, +\begin_inset Formula $aH=Ha\Leftrightarrow H=aHa^{-1}$ +\end_inset + +, +\begin_inset Formula $aH\subseteq G\Leftrightarrow a\in H$ +\end_inset + + +\end_layout + +\begin_layout Standard +general linear +\begin_inset Formula $GL_{2}\mathbb{R}=\left\{ A\in M_{2,2}\mathbb{R},\det A\not=0\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +special linear +\begin_inset Formula $SL_{2}\mathbb{R}=\left\{ A\in M_{2,2}\mathbb{R},\det A=1\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +Lagrange: +\begin_inset Formula $H$ +\end_inset + + podgrupa končne grupe +\begin_inset Formula $G\Rightarrow\left|H\right|\text{ deli }\left|G\right|$ +\end_inset + + in število različnih levih/desnih odeskov po +\begin_inset Formula $H$ +\end_inset + + je +\begin_inset Formula $\frac{\left|G\right|}{\left|H\right|}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $a\in$ +\end_inset + + končne grupe +\begin_inset Formula $G\Rightarrow$ +\end_inset + + red +\begin_inset Formula $a$ +\end_inset + + deli +\begin_inset Formula $\left|G\right|$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Podgrupe edinke in faktorske grupe +\end_layout + +\begin_layout Standard +\begin_inset Formula $H$ +\end_inset + + podgrupa +\begin_inset Formula $G$ +\end_inset + + je edinka +\begin_inset Formula $\Leftrightarrow\forall a\in G:aH=Ha\sim aHa^{-1}=H$ +\end_inset + + +\end_layout + +\begin_layout Standard +Podgrupa konjugiranka +\begin_inset Formula $H^{a}\coloneqq aHa^{-1}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $H\triangleleft G\sim H$ +\end_inset + + edinka v +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $\Leftrightarrow\forall a\in G:H^{A}=H$ +\end_inset + + +\end_layout + +\begin_layout Standard +V Abelovi grupi je vsaka podgrupa edinka. +\end_layout + +\begin_layout Standard +Center je podgrupa edinka. +\end_layout + +\begin_layout Standard +\begin_inset Formula $G/H\coloneqq\left\{ aH:a\in G\right\} $ +\end_inset + + z oper. + +\begin_inset Formula $\left(aH\right)*\left(bH\right)=\left(ab\right)H$ +\end_inset + + +\end_layout + +\begin_layout Standard +Izrek o faktorskih grupah: +\begin_inset Formula $H\triangleleft G\Rightarrow\left(G/H,*\right)$ +\end_inset + + grupa +\end_layout + +\begin_layout Paragraph +Kolobarji in polja +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(R,+,\cdot\right)\text{kolobar}\Rightarrow$ +\end_inset + + +\begin_inset Formula $\left(R,+\right)$ +\end_inset + + abelova grupa, +\begin_inset Formula $\left(R,\cdot\right)$ +\end_inset + + polgrupa, distributivnost. +\end_layout + +\begin_layout Standard +Kolobar je komutativen, če je +\begin_inset Formula $\cdot$ +\end_inset + + komutativna. +\end_layout + +\begin_layout Standard +Kolobar je z enoto, če obstaja enota za +\begin_inset Formula $\cdot$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Direktna vsota kolobarjev +\begin_inset Formula $\left(R\oplus S,+,\cdot\right)$ +\end_inset + + je kolobar. + +\begin_inset Formula $R\oplus S=R\times S$ +\end_inset + +, +\begin_inset Formula $+$ +\end_inset + + in +\begin_inset Formula $\cdot$ +\end_inset + + po komponentah. +\end_layout + +\begin_layout Standard +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $S$ +\end_inset + + komutativna +\begin_inset Formula $\Rightarrow R\oplus S$ +\end_inset + + komutativen. +\end_layout + +\begin_layout Standard +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $S$ +\end_inset + + z enoto +\begin_inset Formula $\Rightarrow R\oplus S$ +\end_inset + + z enoto. +\end_layout + +\begin_layout Standard +\begin_inset Formula $R$ +\end_inset + + kolobar. + +\begin_inset Formula $S\subseteq R$ +\end_inset + + je za podedovani operaciji kolobar, če je +\begin_inset Formula $\left(S,+,\cdot\right)$ +\end_inset + + kolobar. + Izrek: +\begin_inset Formula $S$ +\end_inset + + podkolobar +\begin_inset Formula $\Leftrightarrow0\in S$ +\end_inset + + in zaprta za +\begin_inset Formula $-,\cdot$ +\end_inset + + +\end_layout + +\begin_layout Standard +Center kolobarja: +\begin_inset Formula $ZR=\left\{ a\in R;\forall x\in R:ax=xa\right\} $ +\end_inset + + je podk. +\end_layout + +\begin_layout Paragraph +Delitelji niča in celi kolobarji +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(\mathbb{Z}_{n},+_{n},\cdot_{n}\right)$ +\end_inset + + je vedno kol. +\end_layout + +\begin_layout Standard +Def.: Če v kolobarju velja +\begin_inset Formula $ab=0$ +\end_inset + + in +\begin_inset Formula $a\not=0$ +\end_inset + + in +\begin_inset Formula $b\not=0$ +\end_inset + +, sta +\begin_inset Formula $a$ +\end_inset + + in +\begin_inset Formula $b$ +\end_inset + + delitelja niča. +\end_layout + +\begin_layout Standard +Pravilo krajšanja: +\begin_inset Formula $\forall a,b,c\in R,a\not=0:ab=ac\Rightarrow b=c$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $R$ +\end_inset + + cel +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + komutativen z enoto +\begin_inset Formula $1\not=0$ +\end_inset + + brez deliteljev niča. +\end_layout + +\begin_layout Standard +Za komutativen z enoto +\begin_inset Formula $1\not=0$ +\end_inset + + velja: cel +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + velja prav. + krajš. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + z enoto +\begin_inset Formula $1\not=0$ +\end_inset + + je polje, če je +\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$ +\end_inset + + abelova g. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + z enoto +\begin_inset Formula $1\not=0$ +\end_inset + + je obseg, če je +\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$ +\end_inset + + grupa. +\end_layout + +\begin_layout Standard +Vsako polje je cel kolobar. +\end_layout + +\begin_layout Standard +Polje je cel kolobar z multiplik. + inverzi za vse neničelne. +\end_layout + +\begin_layout Standard +Končen cel kolobar +\begin_inset Formula $\Rightarrow$ +\end_inset + + polje. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall n\in\mathbb{N}\setminus\left\{ 1\right\} :\mathbb{Z}_{n}$ +\end_inset + + cel +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + +\begin_inset Formula $\mathbb{Z}_{n}$ +\end_inset + + polje +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + +\begin_inset Formula $n$ +\end_inset + + praštevilo +\end_layout + +\begin_layout Standard +\begin_inset Formula $S\subseteq$ +\end_inset + + polje +\begin_inset Formula $R$ +\end_inset + + je podpolje za poded. + operaciji, če je +\begin_inset Formula $S$ +\end_inset + + polje. + Zadošča preveriti +\begin_inset Formula $1\in S$ +\end_inset + +, zaprtost za +\begin_inset Formula $-$ +\end_inset + + in deljenje z nenič.ž +\end_layout + +\begin_layout Standard +\begin_inset Formula $n\cdot'a$ +\end_inset + + naj pomeni +\begin_inset Formula $a+\cdots+a$ +\end_inset + + +\begin_inset Formula $n$ +\end_inset + +-krat za +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Karakteristika kolobarja je najmanjši +\begin_inset Formula $n\in N\ni:\forall a\in R:n\cdot'a=0$ +\end_inset + +. + Če ne obstaja, pravimo char +\begin_inset Formula $R=0$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Izrek: Če je red1 +\begin_inset Formula $=n\Rightarrow$ +\end_inset + + char +\begin_inset Formula $R=n$ +\end_inset + + +\end_layout + +\begin_layout Standard +Izrek: Za cel kolobar +\begin_inset Formula $R$ +\end_inset + + je char +\begin_inset Formula $R=0$ +\end_inset + + ali char +\begin_inset Formula $R=$ +\end_inset + + praštevilo +\end_layout + +\begin_layout Paragraph +Ideali +\end_layout + +\begin_layout Standard +\begin_inset Formula $S$ +\end_inset + + podkolobar +\begin_inset Formula $R$ +\end_inset + + je ideal +\begin_inset Formula $R\Leftrightarrow\forall r\in R,s\in S:rs,sr\in S$ +\end_inset + + +\end_layout + +\begin_layout Standard +Primer: večkratniki +\begin_inset Formula $n$ +\end_inset + + so za fiksen +\begin_inset Formula $n$ +\end_inset + + ideal v +\begin_inset Formula $\mathbb{Z}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $I\subseteq R$ +\end_inset + + je ideal +\begin_inset Formula $\Leftrightarrow0\in I$ +\end_inset + +, zaprt za +\begin_inset Formula $-$ +\end_inset + +, zaprt za zunanje množ. +\end_layout + +\begin_layout Standard +Operaciji nad ideali: +\begin_inset Formula $I\overset{+}{\cdot}J=\left\{ i\overset{+}{\cdot}j;\forall i\in I,j\in J\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +Za ideala +\begin_inset Formula $I,J$ +\end_inset + + v +\begin_inset Formula $R$ +\end_inset + + sta +\begin_inset Formula $I+J$ +\end_inset + + in +\begin_inset Formula $I\cdot J$ +\end_inset + + zopet ideala v +\begin_inset Formula $R$ +\end_inset + +. +\end_layout + +\begin_layout Standard +V aditivne odseke +\begin_inset Formula $R/I=\left\{ a+I;\forall a\in R\right\} $ +\end_inset + + vpeljemo operaciji +\begin_inset Formula $\left(a+I\right)\overset{+}{\cdot}\left(b+I\right)=\left(a\overset{+}{\cdot}b\right)I$ +\end_inset + +. + Če je +\begin_inset Formula $I$ +\end_inset + + ideal v +\begin_inset Formula $R$ +\end_inset + +, je +\begin_inset Formula $R/I$ +\end_inset + + za operaciji kolobar. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{multicols} +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/šola/ds2/teor.lyx b/šola/ds2/teor.lyx new file mode 100644 index 0000000..2d03ffc --- /dev/null +++ b/šola/ds2/teor.lyx @@ -0,0 +1,7560 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\begin_preamble +\usepackage{hyperref} +\usepackage{siunitx} +\usepackage{pgfplots} +\usepackage{listings} +\usepackage{multicol} +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}} +\usepackage{amsmath} +\usepackage{tikz} +\newcommand{\udensdash}[1]{% + \tikz[baseline=(todotted.base)]{ + \node[inner sep=1pt,outer sep=0pt] (todotted) {#1}; + \draw[densely dashed] (todotted.south west) -- (todotted.south east); + }% +}% +\DeclareMathOperator{\Lin}{Lin} +\DeclareMathOperator{\rang}{rang} +\DeclareMathOperator{\sled}{sled} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\red}{red} +\DeclareMathOperator{\karakteristika}{char} +\usepackage{algorithm,algpseudocode} +\providecommand{\corollaryname}{Posledica} +\end_preamble +\use_default_options true +\begin_modules +enumitem +theorems-ams +\end_modules +\maintain_unincluded_children false +\language slovene +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification false +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 2cm +\topmargin 2cm +\rightmargin 2cm +\bottommargin 2cm +\headheight 2cm +\headsep 2cm +\footskip 1cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style german +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +Odgovori na vprašanja za teoretični del izpita DS2 IŠRM 2023/24 +\end_layout + +\begin_layout Author + +\noun on +Anton Luka Šijanec +\end_layout + +\begin_layout Date +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +today +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Abstract +Vprašanja je zbral profesor Sandi Klavžar. + Odgovore sem sestavil po svojih zapiskih z njegovih predavanj in drugih + virih. +\end_layout + +\begin_layout Standard +\begin_inset CommandInset toc +LatexCommand tableofcontents + +\end_inset + + +\end_layout + +\begin_layout Section +Slog +\end_layout + +\begin_layout Itemize +Z +\begin_inset Formula $M_{m,n}\mathbb{F}$ +\end_inset + + označim množico matrik z +\begin_inset Formula $m$ +\end_inset + + vrsticami in +\begin_inset Formula $n$ +\end_inset + + stolpci nad poljem +\begin_inset Formula $\mathbb{F}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Znak za množenje ali dvojiški operator v grupi izpuščam. + V bigrupoidih izpuščen operator pomeni množenje (drugo operacijo). +\end_layout + +\begin_layout Section +Vprašanja in odgovori +\end_layout + +\begin_layout Subsection +Kaj je stopnja vozlišča grafa in kaj pravi lema o rokovanju? Kako dokažemo + to lemo? +\end_layout + +\begin_layout Standard +Naj bo +\begin_inset Formula $G$ +\end_inset + + graf. +\end_layout + +\begin_layout Definition* +Stopnja vozlišča grafa +\begin_inset Formula $\deg v$ +\end_inset + + za +\begin_inset Formula $v\in VG$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +Ko eniški operator zahteva en operand, izpustim oklepaje. + Primer: +\begin_inset Formula $\sin x$ +\end_inset + +, +\begin_inset Formula $VG$ +\end_inset + +, +\begin_inset Formula $\chi G$ +\end_inset + +, +\begin_inset Formula $M_{2,2}\mathbb{R}$ +\end_inset + +. +\end_layout + +\end_inset + + predstavlja število povezav, ki imajo to vozlišče kot krajišče. + +\begin_inset Formula $\deg v=\left|\left\{ e\in EG;v\in e\right\} \right|$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +Povezava v grafu je množica natanko dveh vozlišč, toda oklepaje in vejico + izpuščam. + Za +\begin_inset Formula $u,v\in VG$ +\end_inset + + pišem +\begin_inset Formula $uv\in EG$ +\end_inset + +. + Na +\begin_inset Formula $e\in EG$ +\end_inset + + je torej veljavno gledati kot na množico, zato je +\begin_inset Formula $v\in e$ +\end_inset + + smiseln izraz. +\end_layout + +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +Zapis množic: +\begin_inset Formula $\left\{ e\in A;Pe\right\} $ +\end_inset + + pomeni vse take elemente +\begin_inset Formula $A$ +\end_inset + +, za katere velja predikat +\begin_inset Formula $P$ +\end_inset + +. + Ta predikat je običajno izraz. +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Incidenčna matrika +\begin_inset Formula $BG$ +\end_inset + + za +\begin_inset Formula $VG=\left\{ v_{1},\dots,v_{n}\right\} $ +\end_inset + +, +\begin_inset Formula $EG=\left\{ e_{1},\dots,e_{n}\right\} $ +\end_inset + + je široka +\begin_inset Formula $m$ +\end_inset + + in visoka +\begin_inset Formula $n$ +\end_inset + + in velja +\begin_inset Formula $BG_{ij}=\begin{cases} +1 & ;v_{i}\in e_{i}\\ +0 & ;\text{sicer} +\end{cases}$ +\end_inset + +. +\end_layout + +\begin_layout Lemma* +Lema o rokovanju pravi, da je dvakratnik števila povezav enak vsoti vseh + stopenj vozlišč v grafu +\begin_inset Foot +status open + +\begin_layout Plain Layout +Ko je omenjen graf, je običajno mišljen neusmerjen končen graf. +\end_layout + +\end_inset + +, torej +\begin_inset Formula +\[ +\sum_{v\in VG}\deg v=2\left|EG\right|. +\] + +\end_inset + + +\end_layout + +\begin_layout Remark* +Posledica leme o rokovanju je, da je v vsakem grafu sodo vozlišč lihe stopnje, + saj je vsota soda. +\end_layout + +\begin_layout Proof +Če po vozliščih preštevamo povezave, ki se stikajo vozlišča, bi vsako povezavo + šteli dvakrat; vsakič za eno njeno krajišče. + ZDB V vsakem stolpcu incidenčne matrike sta natanko dve enici. + Torej je enic +\begin_inset Formula $2\left|EG\right|$ +\end_inset + +. + V vsaki vrstici incidenčne matrike je toliko enic, kolikor je stopnja +\begin_inset Formula $i-$ +\end_inset + +tega vozlišča, torej je enic +\begin_inset Formula $\sum_{v\in VG}\deg v$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Pojasnite sprehod, sklenjen sprehod, pot v grafu, cikel v grafu. + Pokažite, da vsak graf, ki vsebuje sklenjen sprehod lihe dolžine, vsebuje + tudi cikel lihe dolžine. +\end_layout + +\begin_layout Standard +Naj bo +\begin_inset Formula $G=\left(VG,EG\right)$ +\end_inset + +. +\end_layout + +\begin_layout Definition* +Sprehod v +\begin_inset Formula $G$ +\end_inset + + je zaporedje vozlišč +\begin_inset Formula $v_{0},\dots,v_{k}$ +\end_inset + +, +\begin_inset Formula $k\geq0$ +\end_inset + +, tako da je +\begin_inset Formula $\forall i\in\left\{ 1..k\right\} :v_{i}v_{i+1}\in EG$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +S +\begin_inset Formula $\left\{ k..n\right\} $ +\end_inset + + označujem množico celih števil od vključno +\begin_inset Formula $k$ +\end_inset + + do vključno +\begin_inset Formula $n$ +\end_inset + +, kot to naredi +\family typewriter +bash. + +\family default +Nekateri bi +\begin_inset Formula $\left\{ 1..n\right\} $ +\end_inset + + označili z +\begin_inset Formula $\left[n\right]$ +\end_inset + +. +\end_layout + +\end_inset + +. + Dolžina sprehoda je število prehojenih povezav. + Sprehod je +\series bold +sklenjen +\series default +, če +\begin_inset Formula $v_{0}=v_{k}$ +\end_inset + +. + Sprehod je +\series bold +enostaven +\series default +, če so vsa vozlišča medsebojno različna, z izjemo +\begin_inset Formula $v_{0}$ +\end_inset + + sme biti +\begin_inset Formula $v_{k}$ +\end_inset + +. +\end_layout + +\begin_layout Claim* +Če v grafu +\begin_inset Formula $\exists$ +\end_inset + + sprehod med dvema vozliščema, med njima obstaja tudi enostaven sprehod. +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $Q$ +\end_inset + + sprehod med +\begin_inset Formula $u$ +\end_inset + + in +\begin_inset Formula $v$ +\end_inset + +. + Če je enostaven, je najden, sicer se ponovita vsaj dve vozlišči in je +\begin_inset Formula $Q$ +\end_inset + + oblike +\begin_inset Formula $u,\dots,x,\dots,x,\dots,v$ +\end_inset + +. + Sprehodu priredimo +\begin_inset Formula $Q'$ +\end_inset + +, ki odstrani pot od prvega do zadnjega podvojenega vozlišča +\begin_inset Formula $x$ +\end_inset + +, torej je +\begin_inset Formula $Q'$ +\end_inset + + oblike +\begin_inset Formula $u,\dots,x,\dots,v$ +\end_inset + + (pravzaprav smo odstranili cikel iz poti). + Če je +\begin_inset Formula $Q'$ +\end_inset + + enostaven, je iskani, sicer postopek ponovimo in v končno korakih se postopek + ustavi, saj na vsakem koraku za vsaj 2 zmanjšamo dolžino sicer končno dolgega + sprehoda. +\end_layout + +\begin_layout Definition* +Pot v grafu je podgraf, ki je enostaven sprehod med dvema vozliščema in + je graf Pot ( +\begin_inset Formula $P_{n}$ +\end_inset + +). +\end_layout + +\begin_layout Remark* +Ko govorimo o različnosti/enakosti poti, mislimo različnost/enakost poti + kot podgraf. + Poti sta torej enaki natanko tedaj, ko sta zaporedji (ne množici) vozlišč + poti enaki. +\end_layout + +\begin_layout Definition* +Cikel v grafu je podgraf, ki je enostaven sklenjen sprehod dolžine vsaj + 3. +\end_layout + +\begin_layout Claim* +Če med dvema vozliščema v grafu +\begin_inset Formula $\exists$ +\end_inset + + dve različni poti, potem graf premore cikel. +\end_layout + +\begin_layout Proof +Naj bosta +\begin_inset Formula $P$ +\end_inset + + in +\begin_inset Formula $P'$ +\end_inset + + dve različni poti od +\begin_inset Formula $u$ +\end_inset + + do +\begin_inset Formula $v$ +\end_inset + +. + Naj bo +\begin_inset Formula $x$ +\end_inset + + zadnje (če gledamo usmerjeno +\begin_inset Formula $u,v-$ +\end_inset + +pot) vozlišče, ki je skupno +\begin_inset Formula $P$ +\end_inset + + in +\begin_inset Formula $P'$ +\end_inset + +. + +\begin_inset Formula $x$ +\end_inset + + je lahko +\begin_inset Formula $u$ +\end_inset + +. + Naj bo +\begin_inset Formula $g$ +\end_inset + + prvo naslednje vozlišče za +\begin_inset Formula $x$ +\end_inset + +, ki je na +\begin_inset Formula $P$ +\end_inset + + in +\begin_inset Formula $P'$ +\end_inset + +. + Unija podpoti +\begin_inset Formula $P$ +\end_inset + + od +\begin_inset Formula $x$ +\end_inset + + do +\begin_inset Formula $y$ +\end_inset + + in podpoti +\begin_inset Formula $P'$ +\end_inset + + od +\begin_inset Formula $x$ +\end_inset + + do +\begin_inset Formula $y$ +\end_inset + + določa iskani cikel v grafu. +\end_layout + +\begin_layout Claim* +Če graf premore sklenjen sprehod lihe dolžine, potem premore cikel lihe + dolžine. +\end_layout + +\begin_layout Proof +Indukcija po dolžini sprehoda. + Naj bo +\begin_inset Formula $m$ +\end_inset + + dolžina sprehoda. +\end_layout + +\begin_layout Proof +Baza: +\begin_inset Formula $m=3$ +\end_inset + +, najmanjši sklenjen lihi sprehod je cikel dolžine 3, +\begin_inset Formula $m=4$ +\end_inset + +, drugi najmanjši sklenjen lihi sprehod je cikel dolžine 4. +\end_layout + +\begin_layout Proof +Korak: Naj bo +\begin_inset Formula $Q$ +\end_inset + + poljuben sklenjen sprehod dolžine +\begin_inset Formula $m\geq5$ +\end_inset + +. + Če je +\begin_inset Formula $Q$ +\end_inset + + enostaven, je cikel po definiciji, sicer se vsaj eno vozlišče na sprehodu + vsaj ponovi: +\begin_inset Formula $u,x_{1},\dots,x_{i-1},x_{i},x_{i+1},\dots,x_{j-1},x_{j},x_{j+1},\dots,v$ +\end_inset + + in velja +\begin_inset Formula $x_{i}=x_{j}$ +\end_inset + +. + Poglejmo sprehoda +\begin_inset Formula $Q'=x_{i},\dots,x_{j}=x_{i}$ +\end_inset + + in +\begin_inset Formula $Q''=u,x_{1},\dots,x_{i},x_{j+1},\dots,x_{m}=u$ +\end_inset + +. + +\begin_inset Formula $Q'$ +\end_inset + + in +\begin_inset Formula $Q''$ +\end_inset + + sta sklenjena sprehoda z dolžinama +\begin_inset Formula $m'$ +\end_inset + + in +\begin_inset Formula $m''$ +\end_inset + + in velja +\begin_inset Formula $m=m'+m''$ +\end_inset + +. + Ker je +\begin_inset Formula $m$ +\end_inset + + lih, mora biti lih natanko en izmed +\begin_inset Formula $m'$ +\end_inset + + in +\begin_inset Formula $m''$ +\end_inset + +. + BSŠ +\begin_inset Foot +status open + +\begin_layout Plain Layout +Brez Škode za Splošnost trdimo, da +\end_layout + +\end_inset + + +\begin_inset Formula $m'$ +\end_inset + + lih in +\begin_inset Formula $m'<m$ +\end_inset + +, zato po I. + P. + +\begin_inset Formula $m'$ +\end_inset + + vsebuje lih cikel. +\end_layout + +\begin_layout Subsection +Kaj so dvodelni grafi? Kako jih karakteriziramo? Kako dokažemo to karakterizacij +o? +\end_layout + +\begin_layout Definition* +\begin_inset Formula $G$ +\end_inset + + dvodelen +\begin_inset Foot +status open + +\begin_layout Plain Layout +Pomožni glagol biti med osebkom in povedkovnikom izpuščam. + +\begin_inset Quotes gld +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + dvodelen +\begin_inset Quotes grd +\end_inset + + namesto +\begin_inset Quotes gld +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + je dvodelen +\begin_inset Quotes grd +\end_inset + +. +\end_layout + +\end_inset + + +\begin_inset Formula $\Leftrightarrow\exists A,B\subseteq VG\ni:A\cup B=VG,A\cap B=\emptyset\ni:\forall uv\in EG:u\in A,v\in B\vee v\in A,u\in B$ +\end_inset + +. + ZDB +\begin_inset Foot +status open + +\begin_layout Plain Layout +Z Drugimi Besedami: +\end_layout + +\end_inset + + obstaja razdelitev vozlišč na dve množici, da induciran podgraf posamezne + množice ne vsebuje povezav. + S +\begin_inset Formula $K_{m,n}$ +\end_inset + + označimo poln dvodelni graf +\begin_inset Formula $\left|A\right|=m$ +\end_inset + +, +\begin_inset Formula $\left|B\right|=n$ +\end_inset + +. + Paru +\begin_inset Formula $\left(A,B\right)$ +\end_inset + + pravimo dvodelna razdelitev. +\end_layout + +\begin_layout Theorem* +\begin_inset Formula $G$ +\end_inset + + dvodelen +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + ne vsebuje lihih ciklov. +\end_layout + +\begin_layout Proof +\begin_inset Formula $G$ +\end_inset + + dvodelen +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + vsaka komponenta +\begin_inset Formula $G$ +\end_inset + + dvodelna, zato BSŠ +\begin_inset Formula $G$ +\end_inset + + povezan. +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + Očitno: Če +\begin_inset Formula $G$ +\end_inset + + vsebuje lih cikel, zagotovo ni dvodelen, saj ne moremo razdeliti niti množice + vozlišč cikla. + S skico dokažemo, da sodi cikli so dvodelni, lihi pa niso (narišemo cikel + kot pot v obliki skeletne formule nenasičenega acikličnega alkana in povežemo + prvo in zadnje vozlišče). +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + je po predpostavki brez lihih ciklov ****. + Izberimo poljubno +\begin_inset Formula $x_{0}\in VG$ +\end_inset + +. + Naj bo +\begin_inset Formula $A=\left\{ u\in VG;d_{G}\left(u,x_{0}\right)\text{ sod}\right\} ,B=\left\{ u\in VG;d_{G}\left(u,x_{0}\right)\text{ lih}\right\} $ +\end_inset + +. + +\begin_inset Formula $x_{0}$ +\end_inset + + je torej v +\begin_inset Formula $A$ +\end_inset + +, saj je +\begin_inset Formula $d_{G}\left(x_{0},x_{0}\right)=0$ +\end_inset + +. + Trdimo, da je +\begin_inset Formula $\left(A,B\right)$ +\end_inset + + dvodelna razdelitev +\begin_inset Formula $G$ +\end_inset + +. + Razdelitev je, ker je +\begin_inset Formula $A\cup B=\emptyset$ +\end_inset + + in +\begin_inset Formula $A\cup B=VG$ +\end_inset + +. + Za dvodelnost pa mora veljati +\begin_inset Formula $\forall X\in\left\{ A,B\right\} :\forall u,v\in X:uv\not\in EG$ +\end_inset + +. + Preverimo za splošen fiksen +\begin_inset Formula $X\in\left\{ A,B\right\} $ +\end_inset + +: Naj bosta +\begin_inset Formula $u,v\in X$ +\end_inset + +, BSŠ +\begin_inset Formula $d_{G}\left(x_{0},u\right)>d_{G}\left(x_{0},v\right)$ +\end_inset + + *. + Ločimo dva primera: +\end_layout + +\begin_deeper +\begin_layout Description +\begin_inset Formula $d_{G}\left(x_{0},u\right)\not=d_{G}\left(x_{0},v\right):$ +\end_inset + + Vsled iste parnosti velja +\begin_inset Formula $\left|d_{G}\left(x_{0},u\right)-d_{G}\left(x_{0},v\right)\right|\geq2$ +\end_inset + + ***. + PDDRAA +\begin_inset Foot +status open + +\begin_layout Plain Layout +Pa Denimo Da sledeča trditev ne drži (Reductio Ad Absurdum) +\end_layout + +\end_inset + + +\begin_inset Formula $uv\in EG$ +\end_inset + +, tedaj se +\begin_inset Formula $d_{G}\left(x_{0},u\right)$ +\end_inset + + in +\begin_inset Formula $d_{G}\left(x_{0},v\right)$ +\end_inset + + razlikujeta za največ 1 in velja +\begin_inset Formula $d_{G}\left(x_{0},u\right)\leq d_{G}\left(x_{0},v\right)+1$ +\end_inset + + **. + Iz * in ** sledi +\begin_inset Formula $d_{G}\left(x_{0},u\right)-d_{G}\left(x_{0},v\right)=1$ +\end_inset + +, kar je v +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + + s trditvijo ***. +\end_layout + +\begin_layout Description +\begin_inset Formula $d_{G}\left(x_{0},u\right)=d_{G}\left(x_{0},v\right):$ +\end_inset + + Naj bo +\begin_inset Formula $P_{x}$ +\end_inset + + najkrajša +\begin_inset Formula $x_{0},x-$ +\end_inset + +pot. + PDDRAA +\begin_inset Formula $uv\in EG$ +\end_inset + +, tedaj +\begin_inset Formula $P_{u}=\left\{ ...P_{v},v\right\} $ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +Prefiksni razširitveni operator +\begin_inset Formula $...$ +\end_inset + + razširi množico oziroma v tem primeru zaporedje v seznam elementov. + S tem lahko množico uporabimo kot seznam. + Povzemam operator +\family typewriter +... + +\family default + iz javascripta. +\end_layout + +\end_inset + +, torej +\begin_inset Formula $\left|P_{u}\right|=1+\left|P_{v}\right|$ +\end_inset + +. + Cikel, ki ga tvorijo +\begin_inset Formula $P_{u}$ +\end_inset + +, +\begin_inset Formula $P_{v}$ +\end_inset + + in povezava +\begin_inset Formula $uv$ +\end_inset + +, je torej dolžine +\begin_inset Formula $2\left|P_{v}\right|+1$ +\end_inset + +, kar je liho število, kar je v +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + + s trditvijo ****. +\end_layout + +\end_deeper +\begin_layout Subsection +Kaj je homomorfizem grafov, izomorfizem grafov in avtomorfizem grafa? Kaj + je to +\begin_inset Formula $\Aut\left(G\right)$ +\end_inset + +? Kakšno algebrsko strukturo ima? +\end_layout + +\begin_layout Standard +Naj bosta +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $H$ +\end_inset + + grafa. +\end_layout + +\begin_layout Definition* +Preslikava +\begin_inset Formula $f:VG\to VH$ +\end_inset + + je hm +\begin_inset Formula $\varphi$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +homomorfizem +\end_layout + +\end_inset + + grafov +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $H\Leftrightarrow\forall u,v\in VG:uv\in EG\Rightarrow fufv\in EH$ +\end_inset + +, ZDB če slika povezave v povezave. +\end_layout + +\begin_layout Remark* +Če je +\begin_inset Formula $f$ +\end_inset + + hm +\begin_inset Formula $\text{\ensuremath{\varphi}},$ +\end_inset + +porodi preslikavo +\begin_inset Formula $f':EG\to EH$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $f:VK_{n,m}\to VK_{2}$ +\end_inset + + s predpisom +\begin_inset Formula $fx=\begin{cases} +u & ;x\in A\\ +v & ;x\in B +\end{cases}$ +\end_inset + + je hm +\begin_inset Formula $\varphi$ +\end_inset + +. + +\begin_inset Formula $K_{2}$ +\end_inset + + je homomorfna slika vsakega dvodelnega grafa. +\end_layout + +\begin_layout Definition* +Če je hm +\begin_inset Formula $\varphi$ +\end_inset + + surjektiven po povezavah in vozliščih, je epimorfizem. + Če je injektiven na vozliščih (in posledično na povezavah), je monomorfizem + ali vložitev. + Vložitev je izometrična, če +\begin_inset Formula $\forall u,v\in VG:d_{G}\left(u,v\right)=d_{H}\left(fu,fv\right)$ +\end_inset + + ZDB ohranja razdalje. +\end_layout + +\begin_layout Claim* +Če sta +\begin_inset Formula $f:VG\to VH$ +\end_inset + + in +\begin_inset Formula $g:VH\to VK$ +\end_inset + + hm +\begin_inset Formula $\varphi$ +\end_inset + +, je +\begin_inset Formula $g\circ f:VG\to VK$ +\end_inset + + spet hm +\begin_inset Formula $\varphi$ +\end_inset + +. +\end_layout + +\begin_layout Definition* +Če je +\begin_inset Formula $f:VG\to VH$ +\end_inset + + bijekcija, hm +\begin_inset Formula $\varphi$ +\end_inset + + in +\begin_inset Formula $f^{-1}$ +\end_inset + + hm +\begin_inset Formula $\varphi$ +\end_inset + + (ZDB slika nepovezave v nepovezave), je +\begin_inset Formula $f$ +\end_inset + + im +\begin_inset Formula $\varphi$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +izomorfizem +\end_layout + +\end_inset + + grafov +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $H$ +\end_inset + +. + ZDB +\begin_inset Formula $f:VG\to VH$ +\end_inset + + im +\begin_inset Formula $\varphi$ +\end_inset + + +\begin_inset Formula $\Leftrightarrow f$ +\end_inset + + bijekcija +\begin_inset Formula $\wedge\forall u,v\in VG:uv\in EG\Leftrightarrow fufv\in EH$ +\end_inset + +. + Če +\begin_inset Formula $\text{\ensuremath{\exists}}$ +\end_inset + + im +\begin_inset Formula $\varphi$ +\end_inset + + grafov +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $H$ +\end_inset + +, pravimo, da sta +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $H$ +\end_inset + + izomorfna in pišemo +\begin_inset Formula $G\cong H$ +\end_inset + +. +\end_layout + +\begin_layout Claim* +\begin_inset Formula $\cong$ +\end_inset + + je na +\begin_inset Formula $\mathcal{G}$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +množica vseh grafov +\end_layout + +\end_inset + + ekvivalenčna (refleksivna, simetrična, tranzitivna). +\end_layout + +\begin_layout Definition* +im +\begin_inset Formula $\varphi$ +\end_inset + + +\begin_inset Formula $G\to G$ +\end_inset + + je am +\begin_inset Formula $\varphi$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +avtomorfizem +\end_layout + +\end_inset + +. + Vse am +\begin_inset Formula $\varphi$ +\end_inset + + grafa +\begin_inset Formula $G$ +\end_inset + + združimo v množico in jo opremimo z operacijo komponiranja. + Dobimo +\begin_inset Quotes gld +\end_inset + +grupo avtomorfizmov grafa +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Quotes grd +\end_inset + +, ki jo označimo z +\begin_inset Formula $\Aut G$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\Aut C_{4}=\left(\left\{ id,\left(2,4\right),\left(12\right)\left(34\right),\left(1234\right),\left(13\right),\left(13\right)\left(24\right),\left(1423\right),\left(4321\right)\right\} ,\circ\right)$ +\end_inset + +, +\begin_inset Formula $\Aut K_{n}=\left(S_{n},\circ\right)$ +\end_inset + + za +\begin_inset Formula $S_{n}$ +\end_inset + + množica vseh permutacij +\begin_inset Formula $n$ +\end_inset + + elementov, +\begin_inset Formula $\Aut P_{n}=\left(\left\{ id,f\left(i\right)=n-i-1\right\} ,\circ\right)$ +\end_inset + + +\end_layout + +\begin_layout Fact* +Za vsako končno grupo +\begin_inset Formula $X$ +\end_inset + + +\begin_inset Formula $\exists$ +\end_inset + + graf +\begin_inset Formula $G\ni:\Aut G=X$ +\end_inset + +. + Algebra +\begin_inset Formula $\subseteq$ +\end_inset + + Diskretne strukture. + Dokaz na magisteriju. +\end_layout + +\begin_layout Subsection +Kaj pomeni, da je graf +\begin_inset Formula $H$ +\end_inset + + minor grafa +\begin_inset Formula $G$ +\end_inset + +? Kdaj sta dva grafa homeomorfna? Pojasni operacijo kartezičnega produkta + grafov. +\end_layout + +\begin_layout Definition* + +\series bold +Odstranjevanje vozlišč +\series default +: za neko +\begin_inset Formula $S\subseteq VG$ +\end_inset + + je +\begin_inset Formula $G-S$ +\end_inset + + graf, ki ga dobimo, ko iz +\begin_inset Formula $G$ +\end_inset + + odstranimo vozlišča in vozliščem pripadajoče povezave iz +\begin_inset Formula $S$ +\end_inset + +. + Za +\begin_inset Formula $S=\left\{ u\right\} $ +\end_inset + + pišemo tudi +\begin_inset Formula $G-u$ +\end_inset + +. + +\series bold +Odstranjevanje povezav +\series default +: za neko +\begin_inset Formula $F\subseteq EG$ +\end_inset + + je +\begin_inset Formula $G-F$ +\end_inset + + graf, ki ga dobimo, ko iz +\begin_inset Formula $G$ +\end_inset + + odstranimo povezave iz +\begin_inset Formula $F$ +\end_inset + +. + Za +\begin_inset Formula $S=\left\{ e\right\} $ +\end_inset + + pišemo tudi +\begin_inset Formula $G-e$ +\end_inset + +. + +\series bold +Skrčitev povezave +\series default +: za +\begin_inset Formula $e=\left\{ u,v\right\} \in EG$ +\end_inset + + je +\begin_inset Formula $G/e$ +\end_inset + + graf, ki ga dobimo tako, da identificiramo +\begin_inset Formula $u$ +\end_inset + + in +\begin_inset Formula $v$ +\end_inset + + in odstranimo morebitne vzporedne povezave, s čimer odstranimo tudi zanko + +\begin_inset Formula $\left\{ u,u=v\right\} $ +\end_inset + +. + +\begin_inset Formula $G/e_{1}/e_{2}/\dots/e_{n}$ +\end_inset + + označimo z +\begin_inset Formula $G/\left\{ e_{1},e_{2},\dots,e_{n}\right\} $ +\end_inset + + . +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +\begin_inset Formula $H$ +\end_inset + + minor +\begin_inset Formula $G\Leftrightarrow$ +\end_inset + + +\begin_inset Formula $H$ +\end_inset + + dobimo iz +\begin_inset Formula $G$ +\end_inset + + z nekim zaporedjem operacij, kjer so dovoljene operacije odstranjevanje + vozlišča, odstranjevanje povezave in skrčitev povezave. + Ekvivalentno je +\begin_inset Formula $H$ +\end_inset + + minor +\begin_inset Formula $G$ +\end_inset + +, če ga lahko dobimo iz nekega podgrafa +\begin_inset Formula $G$ +\end_inset + +, ki mu skrčimo poljubno povezav. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Subdivizija povezav: +\begin_inset Formula $G\to G^{+}e$ +\end_inset + + za +\begin_inset Formula $e\in EG$ +\end_inset + +: +\begin_inset Formula $VG^{+}e=VG\cup\left\{ x_{e}\right\} $ +\end_inset + +, +\begin_inset Formula $EG^{+}e=EG\setminus e\cup\left\{ x_{e}u,x_{e}v\right\} $ +\end_inset + +. + Graf +\begin_inset Formula $H$ +\end_inset + + je subdivizija +\begin_inset Formula $G\Leftrightarrow H$ +\end_inset + + dobimo iz +\begin_inset Formula $G$ +\end_inset + + z zaporedjem subdivizij povezav +\begin_inset Formula $G$ +\end_inset + + ZDB povezave v +\begin_inset Formula $G$ +\end_inset + + zamenjamo s poljubno dolčimi potmi. + Relacija je refleksivna ( +\begin_inset Formula $G$ +\end_inset + + subdivizija +\begin_inset Formula $G$ +\end_inset + +). +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Glajenje vozlišča +\begin_inset Formula $u$ +\end_inset + + stopnje +\begin_inset Formula $2$ +\end_inset + +: (obratna operacija od subdivizije) +\begin_inset Formula $G^{-}u=\left(VG\setminus u,\left(EG\setminus\left\{ e,f\right\} \right)\cup\left\{ \left\{ v,w\right\} \right\} \right)$ +\end_inset + +, kjer sta +\begin_inset Formula $e$ +\end_inset + + in +\begin_inset Formula $f$ +\end_inset + + povezavi, ki vsebujeta +\begin_inset Formula $u$ +\end_inset + +, +\begin_inset Formula $v$ +\end_inset + + in +\begin_inset Formula $w$ +\end_inset + + pa njuni drugi krajišči (tisti krajišči, ki nista +\begin_inset Formula $u$ +\end_inset + +). +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Grafa sta homeomorfna, če sta izomorfna po gladitvi vseh vozlišč stopnje + 2. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Naj bosta +\begin_inset Formula $G$ +\end_inset + +, +\begin_inset Formula $H$ +\end_inset + + grafa. + Kartezični produkt grafov +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $H$ +\end_inset + + označimo z +\begin_inset Formula $G\square H$ +\end_inset + +: +\begin_inset Formula $V\left(G\square H\right)=VG\times VH,E\left(G\square H\right)=\left\{ \left(g,h\right)\left(g',h'\right);g=g'\wedge hh'\in EH\vee h=h'\wedge gg'\in EG\right\} $ +\end_inset + +. +\end_layout + +\begin_layout Remark* +\begin_inset Formula $\left(\mathcal{G},\square\right)$ +\end_inset + + je monoid, kajti +\begin_inset Formula $K_{1}$ +\end_inset + + je enota, operacija je komutativna in notranja. +\end_layout + +\begin_layout Example* +\begin_inset Formula $K_{2}\square K_{2}\cong C_{4}$ +\end_inset + + +\end_layout + +\begin_layout Subsection +Kaj so to prerezna vozlišča in prerezne povezave grafa? Kdaj je graf +\begin_inset Formula $k-$ +\end_inset + +povezan in kaj je to povezanost grafa? +\end_layout + +\begin_layout Standard +Naj bo +\begin_inset Formula $G$ +\end_inset + + graf z +\begin_inset Formula $m$ +\end_inset + + vozlišči. +\end_layout + +\begin_layout Definition* +\begin_inset Formula $u,v\in VG$ +\end_inset + + sta v isti povezani komponenti, če v +\begin_inset Formula $G$ +\end_inset + + obstaja sprehod med njima. + Število komponent +\begin_inset Formula $G$ +\end_inset + + označimo z +\begin_inset Formula $\Omega G$ +\end_inset + +. + +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\Leftrightarrow\Omega G=1$ +\end_inset + +. +\end_layout + +\begin_layout Remark* +Biti v isti komponenti je ekvivalenčna relacija (refleksivna, simetrična + in tranzitivna). +\end_layout + +\begin_layout Definition* +\begin_inset Formula $x\in VG$ +\end_inset + + prerezno +\begin_inset Formula $\Leftrightarrow\Omega\left(G-x\right)>\Omega G$ +\end_inset + +. + +\begin_inset Formula $e\in EG$ +\end_inset + + prerezna +\begin_inset Formula $\sim e$ +\end_inset + + most +\begin_inset Formula $\Leftrightarrow\Omega\left(G-e\right)>\Omega G$ +\end_inset + +. + +\begin_inset Formula $X\subseteq VG$ +\end_inset + + je prerezna množica +\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$ +\end_inset + +. + +\begin_inset Formula $X\subseteq EG$ +\end_inset + + je prerezna množica povezav +\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Povezan graf +\begin_inset Formula $G$ +\end_inset + + je +\begin_inset Formula $k-$ +\end_inset + +povezan +\begin_inset Formula $\Leftrightarrow\left|VG\right|\geq k+1\wedge\forall X\text{ prerezna}\subseteq VG:\left|X\right|>=k$ +\end_inset + + prerezna. + ZDB ima vsaj +\begin_inset Formula $k+1$ +\end_inset + + vozlišč in +\series bold +ne +\series default + vsebuje prerezne množice moči manjše od +\begin_inset Formula $k$ +\end_inset + +. + Povezanost grafa +\begin_inset Formula $G\sim\kappa G$ +\end_inset + + je največji +\begin_inset Formula $k\in\mathbb{N}$ +\end_inset + +, za katerega je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +povezan ZDB najmanjše število vozlišč, ki jih moramo odstraniti iz grafa, + da graf ne bo več povezan. +\end_layout + +\begin_layout Remark* +\begin_inset Formula $\kappa G=k\Rightarrow G$ +\end_inset + + nima prerezne množice moči +\begin_inset Formula $<k$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\kappa K_{n}=n-1$ +\end_inset + +, +\begin_inset Formula $\kappa P_{n}=1$ +\end_inset + +, +\begin_inset Formula $\kappa C_{n}=2$ +\end_inset + +, +\begin_inset Formula $\kappa K_{m,n}=\min\left\{ n,m\right\} $ +\end_inset + +, +\begin_inset Formula $\kappa Q_{n}=n$ +\end_inset + +, +\begin_inset Formula $\kappa G\leq\delta G$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +\begin_inset Formula $\delta G$ +\end_inset + + je minimalna stopnja vozlišča v grafu +\begin_inset Formula $G$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Pojasnite Whitney-ev izrek, ki katekterizira +\begin_inset Formula $2-$ +\end_inset + +povezane grafe. + Skicirajte dokaz tega izreka. + Zapišite Mengerjev izrek. +\end_layout + +\begin_layout Definition* +Poti +\begin_inset Formula $\left[p_{1},p_{2},\dots,p_{n-1},p\right]$ +\end_inset + + in +\begin_inset Formula $\left[r_{1},r_{2},\dots,r_{m-1},r_{m}\right]$ +\end_inset + + sta notranje disjunktni, če sta disjunktni množici njunih vozlišč izvzemši + prvo in zadnje vozlišče. +\end_layout + +\begin_layout Theorem* +Graf +\begin_inset Formula $G$ +\end_inset + +, +\begin_inset Formula $\left|VG\right|\geq2$ +\end_inset + +, je +\begin_inset Formula $2-$ +\end_inset + +povezan +\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists$ +\end_inset + + notranje disjunktni +\begin_inset Formula $u,v-$ +\end_inset + +poti. +\end_layout + +\begin_layout Proof +Dokazujemo ekvivalenco: +\end_layout + +\begin_deeper +\begin_layout Description +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + + Po predpostavki za poljubna +\begin_inset Formula $u,v$ +\end_inset + + obstajata dve notranje disjunktni +\begin_inset Formula $u,v-$ +\end_inset + +poti. + Dokazujemo, da je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $2-$ +\end_inset + +povezan +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + nima prereznega vozlišča +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + ne obstaja prerezna množica, ki je singleton +\begin_inset Foot +status open + +\begin_layout Plain Layout +množica moči ena +\end_layout + +\end_inset + +. + Dokažimo, da poljubno +\begin_inset Formula $x\in VG$ +\end_inset + + ni prerezno, torej da po odstranitvi tega vozlišča še vedno obstaja povezava + med poljubnima +\begin_inset Formula $u,v\in V\left(G-x\right)$ +\end_inset + +. + Ločimo 3 primere: +\begin_inset Formula $x$ +\end_inset + + je na prvi +\begin_inset Formula $u,v-$ +\end_inset + +poti, +\begin_inset Formula $x$ +\end_inset + + je na drugi +\begin_inset Formula $u,v-$ +\end_inset + +poti, +\begin_inset Formula $x$ +\end_inset + + ni na nobeni izmed +\begin_inset Formula $u,v-$ +\end_inset + +poti. + V vsakem primeru sta +\begin_inset Formula $u,v$ +\end_inset + + v +\begin_inset Formula $G-x$ +\end_inset + + še vedno v isti povezani komponenti. + +\end_layout + +\begin_layout Description +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + Po predpostavki je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $2-$ +\end_inset + +povezan. + Vzemimo poljubna +\begin_inset Formula $u,v\in VG$ +\end_inset + +. + Indukcija po +\begin_inset Formula $d_{G}\left(u,v\right)$ +\end_inset + +: +\end_layout + +\begin_deeper +\begin_layout Standard +Baza: +\begin_inset Formula $d_{G}\left(u,v\right)=1$ +\end_inset + + (sosednji vozlišči) +\begin_inset Formula $G-e$ +\end_inset + + je povezan, sicer je RAAPDD +\begin_inset Formula $uv$ +\end_inset + + most, kar bi bilo v +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + + s predpostavko, ker bi bil en izmed +\begin_inset Formula $u,v$ +\end_inset + + prerezno, saj ima +\begin_inset Formula $G$ +\end_inset + + vsaj 3 vozlišča in ima vsaj eden izmed +\begin_inset Formula $u,v$ +\end_inset + + še enega soseda. + Sedaj vemo, da +\begin_inset Formula $\exists u,v-$ +\end_inset + +pot, ki ni +\begin_inset Formula $uv$ +\end_inset + +. + Imamo torej dve notranje disjunktni +\begin_inset Formula $u,v-$ +\end_inset + +poti: +\begin_inset Formula $e$ +\end_inset + + in tisto drugo. +\end_layout + +\begin_layout Standard +Korak: +\begin_inset Formula $d_{G}\left(u,v\right)=k\geq2$ +\end_inset + +. + Naj bo +\begin_inset Formula $P$ +\end_inset + + najkrajša +\begin_inset Formula $u,v-$ +\end_inset + +pot. + Predzadnje vozlišče na njej (tik pred +\begin_inset Formula $v$ +\end_inset + +) naj bo +\begin_inset Formula $w$ +\end_inset + +. + +\begin_inset Formula $d_{G}\left(u,w\right)=k-1$ +\end_inset + + in po I. + P. + obstajata dve notranje disjunktni +\begin_inset Formula $u,w-$ +\end_inset + +poti +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $S$ +\end_inset + +. + Ločimo primera: +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $v\in VR\cup VS$ +\end_inset + + Tedaj je +\begin_inset Formula $v$ +\end_inset + + na ciklu, ki ga tvorita poti +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $S$ +\end_inset + + (je na eni izmed poti). + Zato obstajata dve notranje disjunktni +\begin_inset Formula $u,v-$ +\end_inset + +poti; ena v eno smer po ciklu, druga v drugo. +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $v\not\in VR\cup VS$ +\end_inset + + Tedaj +\begin_inset Formula $v$ +\end_inset + + ni na tem ciklu, vendar je sosed +\begin_inset Formula $w$ +\end_inset + +, ki je na ciklu. + +\begin_inset Formula $G-w$ +\end_inset + + mora biti povezan, saj smo odstranili le eno vozlišče, torej +\begin_inset Formula $\exists u,v-$ +\end_inset + +pot +\begin_inset Formula $T$ +\end_inset + +. + Ločimo primera: +\end_layout + +\begin_deeper +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $T\cap\left(VS\cup VR\right)=\left\{ u\right\} $ +\end_inset + + Našli smo dve notranje disjunktni poti v +\begin_inset Formula $G$ +\end_inset + +: +\begin_inset Formula $T$ +\end_inset + + in +\begin_inset Formula $\left[\dots R,v\right]$ +\end_inset + + ( +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $wv$ +\end_inset + + povezava). +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left|T\cap\left(VS\cup VR\right)\right|\geq2$ +\end_inset + + Naj bo +\begin_inset Formula $x$ +\end_inset + + zadnje (kjer je +\begin_inset Formula $v$ +\end_inset + + konec poti) vozlišče na +\begin_inset Formula $T$ +\end_inset + +, ki je na +\begin_inset Formula $R\cup S$ +\end_inset + +. + BSŠ +\begin_inset Formula $x\in VS$ +\end_inset + +. + +\begin_inset Formula $x\not=w$ +\end_inset + + po konstrukciji +\begin_inset Formula $T$ +\end_inset + +. + Dve poti: po +\begin_inset Formula $S$ +\end_inset + + do +\begin_inset Formula $x$ +\end_inset + + in nato po +\begin_inset Formula $T$ +\end_inset + + do +\begin_inset Formula $v$ +\end_inset + + ter +\begin_inset Formula $\left[\dots R,v\right]$ +\end_inset + + ( +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $wv$ +\end_inset + + povezava). +\end_layout + +\end_deeper +\end_deeper +\end_deeper +\begin_layout Theorem* +Menger (posplošitev Whitneya): naj bo +\begin_inset Formula $G$ +\end_inset + + graf z vsak +\begin_inset Formula $k+1$ +\end_inset + + vozlišči. + Tedaj je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +povezan +\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$ +\end_inset + + paroma notranje disjunktnih +\begin_inset Formula $u,v-$ +\end_inset + +poti. +\end_layout + +\begin_layout Definition* +Graf je +\begin_inset Formula $k-$ +\end_inset + + povezan po povezavah, če +\begin_inset Formula $\nexists$ +\end_inset + + prerezna množica povezav moči +\begin_inset Formula $<k$ +\end_inset + +. + Povezanost grafa +\begin_inset Formula $G$ +\end_inset + + po povezavah ( +\begin_inset Formula $\kappa'G$ +\end_inset + +) je največji +\begin_inset Formula $k$ +\end_inset + +, da je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +povezan po povezavah. +\end_layout + +\begin_layout Theorem* +Menger': +\begin_inset Formula $G$ +\end_inset + + je +\begin_inset Formula $k-$ +\end_inset + +povezan po povezavah +\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$ +\end_inset + + po povezavah notranje disjunktnih +\begin_inset Formula $u,v-$ +\end_inset + +poti. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Theorem* +\begin_inset Formula $\forall G\in\mathcal{G}:\kappa G\leq\kappa'G$ +\end_inset + + in +\begin_inset Formula $\kappa'G\leq\delta G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Dokaze zadnjih treh izrekov najdete na magisteriju. +\end_layout + +\begin_layout Subsection +Kaj je drevo in kaj je gozd? Katere katekterizacije dreves poznate? +\end_layout + +\begin_layout Definition* +Gozd je graf brez ciklov. + Drevo je povezan gozd. + List je vozlišče stopnje 1. +\end_layout + +\begin_layout Lemma +\begin_inset CommandInset label +LatexCommand label +name "lem:Vsako-drevo-z" + +\end_inset + +Vsako drevo z vsaj dvema vozliščema premore list. +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $T$ +\end_inset + + drevo, +\begin_inset Formula $v\in VT$ +\end_inset + + poljubno. + +\begin_inset Formula $\left|VT\right|\geq2\wedge T$ +\end_inset + + povezan +\begin_inset Formula $\Rightarrow v$ +\end_inset + + ima soseda +\begin_inset Formula $v_{1}$ +\end_inset + +. + Če je +\begin_inset Formula $v_{1}$ +\end_inset + + edini sosed +\begin_inset Formula $v$ +\end_inset + +, je +\begin_inset Formula $v$ +\end_inset + + list, sicer je tudi +\begin_inset Formula $v_{2}\not=v_{1}$ +\end_inset + + sosed +\begin_inset Formula $v$ +\end_inset + +. + Če je +\begin_inset Formula $v$ +\end_inset + + edini sosed +\begin_inset Formula $v_{2}$ +\end_inset + +, je +\begin_inset Formula $v_{2}$ +\end_inset + + list, sicer je tudi +\begin_inset Formula $v_{3}$ +\end_inset + + sosed +\begin_inset Formula $v$ +\end_inset + +. + Postopek ponavljamo, dokler ne najdemo lista. + V grafu ni ciklov in graf je končen, zato se postopek ustavi. +\end_layout + +\begin_layout Lemma +\begin_inset CommandInset label +LatexCommand label +name "lem:-drevo" + +\end_inset + + +\begin_inset Formula $T$ +\end_inset + + drevo +\begin_inset Formula $\Rightarrow\left|ET\right|=\left|VT\right|-1$ +\end_inset + + +\end_layout + +\begin_layout Proof +z indukcijo po številu vozlišč: +\end_layout + +\begin_layout Proof +Baza: +\begin_inset Formula $\left|VT=1\right|$ +\end_inset + +, +\begin_inset Formula $\left|EG\right|=0$ +\end_inset + + (izolirano vozlišče je drevo) +\end_layout + +\begin_layout Proof +Korak: +\begin_inset Formula $T$ +\end_inset + + poljubno drevo +\begin_inset Formula $\left|VG\right|\geq2$ +\end_inset + +. + +\begin_inset Formula $v\in VT$ +\end_inset + + naj bo list v +\begin_inset Formula $T$ +\end_inset + + po lemi +\begin_inset CommandInset ref +LatexCommand ref +reference "lem:Vsako-drevo-z" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + Po I. + P. + je +\begin_inset Formula $\left|E\left(T-v\right)\right|=\left|V\left(T-v\right)\right|-1$ +\end_inset + +, po konstrukciji +\begin_inset Formula $T-v$ +\end_inset + + pa je +\begin_inset Formula $\left|E\left(T-v\right)\right|=\left|ET\right|-1$ +\end_inset + +, +\begin_inset Formula $\left|V\left(T-v\right)\right|=\left|VT\right|-1$ +\end_inset + +, torej +\begin_inset Formula $\left|ET\right|=\left|VT\right|-1$ +\end_inset + +. +\end_layout + +\begin_layout Lemma +\begin_inset CommandInset label +LatexCommand label +name "lem:lema3drevesa-enaciklu-g-e-povezan" + +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + povezan, +\begin_inset Formula $e\in EG$ +\end_inset + + leži na ciklu +\begin_inset Formula $\Rightarrow G-e$ +\end_inset + + povezan. +\end_layout + +\begin_layout Proof +Trdimo, da v +\begin_inset Formula $G-e\exists u,v-$ +\end_inset + +pot. + Ker je +\begin_inset Formula $G$ +\end_inset + + povezan, +\begin_inset Formula $\exists u,v-$ +\end_inset + +pot +\begin_inset Formula $P$ +\end_inset + + v +\begin_inset Formula $G$ +\end_inset + +. + Če +\begin_inset Formula $e\not\in P$ +\end_inset + +, ta pot obstaja tudi v +\begin_inset Formula $G-e$ +\end_inset + +. + Če +\begin_inset Formula $e\in P$ +\end_inset + +, očitno dobimo pot tako, da +\begin_inset Formula $e$ +\end_inset + + v +\begin_inset Formula $P$ +\end_inset + + nadomestimo s preostankom cikla, na katerem leži +\begin_inset Formula $e$ +\end_inset + + v +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Lemma +\begin_inset CommandInset label +LatexCommand label +name "lem:-povezan-." + +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\Rightarrow\left|EG\right|\geq\left|VG\right|-1$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Če je +\begin_inset Formula $G$ +\end_inset + + drevo, velja enakost po lemi +\begin_inset CommandInset ref +LatexCommand ref +reference "lem:-drevo" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, sicer +\begin_inset Formula $G$ +\end_inset + + premore cikel. + Po lemi +\begin_inset CommandInset ref +LatexCommand ref +reference "lem:lema3drevesa-enaciklu-g-e-povezan" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + +\begin_inset Formula $\exists e\in EG\ni:G-e$ +\end_inset + + povezan. + Odstranjevanje povezav iz ciklov ponavljamo, dokler ne dobimo drevesa +\begin_inset Formula $T$ +\end_inset + +. + Velja +\begin_inset Formula $VT=VG$ +\end_inset + + (*) in +\begin_inset Formula $\left|ET\right|<\left|EG\right|$ +\end_inset + + (**). + Torej: +\begin_inset Formula $\left|VG\right|-1\overset{\text{(*)}}{=}\left|VT\right|-1\overset{\text{lema \ref{lem:-drevo}}}{=}\left|ET\right|\overset{\text{(**)}}{<}\left|EG\right|$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Karakterizacija dreves. + NTSE za graf +\begin_inset Formula $G$ +\end_inset + +: +\end_layout + +\begin_deeper +\begin_layout Enumerate +\begin_inset CommandInset label +LatexCommand label +name "enu:-drevo" + +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + drevo +\end_layout + +\begin_layout Enumerate +\begin_inset CommandInset label +LatexCommand label +name "enu:-pot-ZDB" + +\end_inset + + +\begin_inset Formula $\forall u,v\in VG\exists!$ +\end_inset + + +\begin_inset Formula $u,v-$ +\end_inset + +pot ZDB za vsak par vozlišč obstaja enolična pot +\end_layout + +\begin_layout Enumerate +\begin_inset CommandInset label +LatexCommand label +name "enu:-povezan-" + +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\wedge\forall e\in EG:e$ +\end_inset + + most +\end_layout + +\begin_layout Enumerate +\begin_inset CommandInset label +LatexCommand label +name "enu:-povezan" + +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\wedge\left|EG\right|=\left|VG\right|-1$ +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Proof +Dokazujemo ekvivalenco: +\end_layout + +\begin_deeper +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\ref{enu:-drevo}\Rightarrow\ref{enu:-pot-ZDB}\right)$ +\end_inset + + PDDRAA +\begin_inset Formula $\exists$ +\end_inset + + dve različni +\begin_inset Formula $u,v-$ +\end_inset + +poti za neka +\begin_inset Formula $u,v\in VG$ +\end_inset + +. + Tedaj graf premore cikel +\begin_inset Formula $\Rightarrow$ +\end_inset + + ni gozd +\begin_inset Formula $\Rightarrow$ +\end_inset + + ni drevo +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +. + PDDRAA +\begin_inset Formula $\nexists$ +\end_inset + + +\begin_inset Formula $u,v-$ +\end_inset + +pot za neka +\begin_inset Formula $u,v\in VG\Rightarrow\Omega G\not=1\Rightarrow$ +\end_inset + + ni drevo +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +. +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\ref{enu:-pot-ZDB}\Rightarrow\ref{enu:-povezan-}\right)$ +\end_inset + + PDDRAA +\begin_inset Formula $\exists uv\in EG\ni:$ +\end_inset + + ni most +\begin_inset Formula $\Rightarrow\exists u,v-$ +\end_inset + +pot v +\begin_inset Formula $G-e\Rightarrow\exists$ +\end_inset + + dve različni +\begin_inset Formula $u,v-$ +\end_inset + +poti v +\begin_inset Formula $G$ +\end_inset + + ( +\begin_inset Formula $uv$ +\end_inset + + in tista v +\begin_inset Formula $G-e$ +\end_inset + +) +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +. +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\ref{enu:-povezan-}\Rightarrow\ref{enu:-povezan}\right)$ +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\Rightarrow G$ +\end_inset + + povezan velja vedno. + Dokažimo +\begin_inset Formula $\left|EG\right|=\left|VG\right|-1$ +\end_inset + + z indukcijo po številu vozlišč: +\end_layout + +\begin_deeper +\begin_layout Standard +Baza: V +\begin_inset Formula $K_{2}$ +\end_inset + + je edina povezava most in +\begin_inset Formula $\left|EK_{2}\right|=2-1=1=\left|VK_{2}\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +Korak: +\begin_inset Formula $e\in EG$ +\end_inset + + poljuben: +\begin_inset Formula $\Omega\left(G-e\right)=2$ +\end_inset + +, dve komponenti +\begin_inset Formula $G-e$ +\end_inset + + naj bosta +\begin_inset Formula $G_{1}$ +\end_inset + + in +\begin_inset Formula $G_{2}$ +\end_inset + +. + Slednja sta povezana in za njiju velja, da je vsaka povezava most in sta + manjša grafa. + Po I. + P. + velja +\begin_inset Formula $\left|EG_{1}\right|=\left|VG_{1}\right|-1$ +\end_inset + + in +\begin_inset Formula $\left|EG_{2}\right|=\left|VG_{2}\right|-1$ +\end_inset + +. + Velja +\begin_inset Formula $\left|EG\right|=\left|EG_{1}\right|+\left|EG_{2}\right|+1=\left|VG_{1}\right|-1+\left|VG_{2}\right|-1+1=\left|VG_{1}\right|+\left|VG_{2}\right|-1=\left|VG\right|-1$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\ref{enu:-povezan}\Rightarrow\ref{enu:-pot-ZDB}\right)$ +\end_inset + + PDDRAA +\begin_inset Formula $G$ +\end_inset + + premore cikel +\begin_inset Formula $\overset{\text{lema \ref{lem:lema3drevesa-enaciklu-g-e-povezan}}}{\Rightarrow}$ +\end_inset + + če mu odstranimo povezavo s cikla, bo ostal povezan. + Obenem zaradi velja +\begin_inset Formula $\left|E\left(G-e\right)\right|=\left|V\left(G-e\right)\right|-2$ +\end_inset + +, kar je v +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + + z lemo +\begin_inset CommandInset ref +LatexCommand ref +reference "lem:-povezan-." +plural "false" +caps "false" +noprefix "false" + +\end_inset + + ( +\begin_inset Formula $G$ +\end_inset + + povezan, toda ne velja implikacija). +\end_layout + +\end_deeper +\begin_layout Subsection +Kaj je vpeto drevo grafa? Kateri grafi premorejo vpeta drevesa? Kako lahko + rekurzivno določimo število vpetih dreves povezanega grafa? +\end_layout + +\begin_layout Definition* +Podgraf +\begin_inset Formula $H$ +\end_inset + + grafa +\begin_inset Formula $G$ +\end_inset + + je vpet, če velja +\begin_inset Formula $VP=VG$ +\end_inset + +. + (Lahko pa ima ima +\begin_inset Formula $G$ +\end_inset + + povezave, ki jih +\begin_inset Formula $H$ +\end_inset + + nima). +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Vpeto drevo grafa +\begin_inset Formula $G$ +\end_inset + + je vpet podgraf, ki je drevo. +\end_layout + +\begin_layout Theorem* +\begin_inset Formula $G$ +\end_inset + + povezan +\begin_inset Formula $\Leftrightarrow G$ +\end_inset + + premore vpeto drevo. +\end_layout + +\begin_layout Proof +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + + očitno. + +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + +: Če je +\begin_inset Formula $G$ +\end_inset + + drevo, je sam sebi vpeto drevo, sicer vsebuje cikel in iterativno uporabljamo + lemo +\begin_inset CommandInset ref +LatexCommand ref +reference "lem:lema3drevesa-enaciklu-g-e-povezan" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, dokler ne konstruiramo vpetega drevesa. +\end_layout + +\begin_layout Definition* +\begin_inset Formula $\tau G$ +\end_inset + + naj bo število vpetih dreves grafa +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Remark* +\begin_inset Formula $T$ +\end_inset + + drevo +\begin_inset Formula $\Rightarrow\tau T=1$ +\end_inset + +, +\begin_inset Formula $\Omega G>1\Rightarrow\tau G=0$ +\end_inset + +, +\begin_inset Formula $\tau C_{n}=n$ +\end_inset + + +\end_layout + +\begin_layout Definition* +Z +\begin_inset Formula $G/e$ +\end_inset + + označimo skrčitev povezave +\begin_inset Formula $e$ +\end_inset + + v multigrafu +\begin_inset Foot +status open + +\begin_layout Plain Layout +V multigrafu se ista povezava lahko pojavi večkrat. +\end_layout + +\end_inset + + +\begin_inset Formula $G$ +\end_inset + +. + To je enako kot skrčitev povezave v grafu ( +\begin_inset Formula $G\backslash e$ +\end_inset + +), le da ne odstranimo večkratnih vzporednih povezav. +\end_layout + +\begin_layout Claim* +\begin_inset Formula $G$ +\end_inset + + povezan, +\begin_inset Formula $e\in EG\Rightarrow\tau G=\tau\left(G-e\right)+\tau\left(G/e\right)$ +\end_inset + + +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $T$ +\end_inset + + vpeto drevo v +\begin_inset Formula $G$ +\end_inset + +. + Naj bo +\begin_inset Formula $e\in EG$ +\end_inset + + poljuben fiksen. + Vpetih dreves, za katere +\begin_inset Formula $e\not\in T$ +\end_inset + +, je +\begin_inset Formula $\tau\left(G-e\right)$ +\end_inset + +. + Vpetih dreves, za katere +\begin_inset Formula $e\in T$ +\end_inset + +, je +\begin_inset Formula $\tau\left(G/e\right)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Rekurzivno torej določamo število vpetih dreves grafa z zgornjo trditvijo + tako, da izbiramo poljubne povezave in jih ožamo ter odstranjujemo in nato + seštejemo +\begin_inset Formula $\tau$ +\end_inset + + teh dveh operacij. +\end_layout + +\begin_layout Subsection +Kaj je Laplaceova matrika multigrafa? Kaj pravi Kirchoffov izrek o številu + vpetih dreves multigrafa? +\end_layout + +\begin_layout Definition* +Laplaceova matrika +\begin_inset Formula $LG$ +\end_inset + + je kvadratna matrika dimenzije +\begin_inset Formula $n$ +\end_inset + +, katere vrstice in stolpci so indeksirani z vozlišči multigrafa +\begin_inset Formula $G$ +\end_inset + + in velja +\begin_inset Formula +\[ +LG_{ij}=\begin{cases} +\deg_{G}v & ;i=j\\ +-\left(\text{število povezav med \ensuremath{v_{i}} in \ensuremath{v_{j}}}\right) & ;i\not=j +\end{cases} +\] + +\end_inset + + +\end_layout + +\begin_layout Theorem* +Kirchoff: Če je +\begin_inset Formula $G$ +\end_inset + + povezan multigraf, potem je +\begin_inset Formula $\tau G=\det\left(LG\text{ brez \ensuremath{i}tega stolpca in \ensuremath{i}te vrstice}\right)$ +\end_inset + + za poljuben +\begin_inset Formula $i$ +\end_inset + +. + ZDB +\begin_inset Formula $\tau G=\det LG_{i,i}$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Kaj pomeni, da je graf Eulerjev? Kako karakteriziramo Eulerjeve grafe? Skicirajt +e dokaz slednjega rezultata. +\end_layout + +\begin_layout Definition* +Sprehod v multigrafu je Eulerjev, če vsebuje vse povezave in sicer vsako + zgolj enkrat. + +\series bold +Sklenjen +\series default + Eulerjev sprehod je Eulerjev obhod. + Graf je Eulerjev, če premore Eulerjev obhod. +\end_layout + +\begin_layout Theorem* +\begin_inset Formula $G$ +\end_inset + + Eulerjev +\begin_inset Formula $\Leftrightarrow\forall v\in VG:\deg_{G}v$ +\end_inset + + sod. +\end_layout + +\begin_layout Proof +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + očitno. + +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + +: +\begin_inset Formula $\forall v\in VG:\deg_{G}v$ +\end_inset + + sod +\begin_inset Formula $\Rightarrow$ +\end_inset + + +\begin_inset Formula $G$ +\end_inset + + premore cikel. + Indukcija po številu povezav: +\end_layout + +\begin_layout Proof +Baza: Izolirano vozlišče, multigraf +\begin_inset Formula $VG=\left\{ u,v\right\} ,EG=\left\{ uv,uv\right\} $ +\end_inset + + in +\begin_inset Formula $C_{3}$ +\end_inset + + so vsi Eulerjevi. +\end_layout + +\begin_layout Proof +Korak: Naj bo +\begin_inset Formula $\left|VG\right|\geq4$ +\end_inset + +. + +\begin_inset Formula $G$ +\end_inset + + gotovo premore cikel +\begin_inset Formula $C$ +\end_inset + +, ker je povezan in nima listov (ni drevo). + Naj bo +\begin_inset Formula $H\coloneqq G-EC$ +\end_inset + +. + +\begin_inset Formula $\forall u\in VC:\deg_{H}u=\deg_{G}u-2$ +\end_inset + +, +\begin_inset Formula $\forall u\not\in VC:\deg_{H}u=\deg_{G}u$ +\end_inset + + +\begin_inset Formula $\Rightarrow\forall v\in H:\deg_{H}v$ +\end_inset + + sod +\begin_inset Formula $\Rightarrow$ +\end_inset + + vsaka komponenta +\begin_inset Formula $H$ +\end_inset + + je Eulerjev graf po I. + P., saj je manjši graf. + +\begin_inset Formula $G$ +\end_inset + + je tudi Eulerjev, obhodimo ga lahko po ciklu +\begin_inset Formula $C$ +\end_inset + +; vsakič, ko naletimo na vozlišče, ki je del komponente +\begin_inset Formula $H$ +\end_inset + +, jo obhodimo in nadaljujemo pot po ciklu. +\end_layout + +\begin_layout Paragraph* +Fleuryjev algoritem +\end_layout + +\begin_layout Standard +sprejme graf in vrne obhod +\end_layout + +\begin_layout Enumerate +Začnemo v poljubni povezavi +\end_layout + +\begin_layout Enumerate +Ko povezavo prehodimo, jo izbrišemo +\end_layout + +\begin_layout Enumerate +Postopek nadaljujemo in pri tem pazimo le na to, da gremo na most le v primeru, + če ni druge možnosti. +\end_layout + +\begin_layout Theorem* +Če je +\begin_inset Formula $G$ +\end_inset + + Eulerjev graf, potem Fleuryjev algoritem vrne Eulerjev obhod. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +dodaj dekompozicijo in dekompozicijo v cikle +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Kdaj je graf Hamiltonov? Navedite in pojasnite potrebni pogoj z razpadom + grafa za obstoj Hamiltonovega cikla v grafu. +\end_layout + +\begin_layout Definition* + +\series bold +Hamiltonov cikel +\series default +v grafu je cikel, ki vsebuje vsa vozlišča grafa ZDB Hamiltonov cikel je + vpet podgraf, ki je cikel. + +\series bold +Graf je Hamiltonov +\series default +, če premore Hamiltonov cikel. + +\series bold +Hamiltonova pot +\series default + v grafu je pot, ki vsebuje vsa vozlišča grafa ZDB vpet podgraf, ki je pot. +\end_layout + +\begin_layout Theorem* +\begin_inset Formula $G$ +\end_inset + + Hamiltonov, +\begin_inset Formula $S\subseteq VG\Rightarrow\Omega\left(G-S\right)\leq\left|S\right|$ +\end_inset + +. + (potreben pogoj za obstoj Hamiltonovega cikla) +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $VG=\left\{ v_{1},\dots,v_{n}\right\} $ +\end_inset + + Hamiltonov. + BSŠ naj bo +\begin_inset Formula $\left[v_{1},\dots,v_{n}\right]$ +\end_inset + + Hamiltonov cikel. + Skica dokaza: Naj bosta +\begin_inset Formula $v_{i},v_{j}\in S,i<j$ +\end_inset + +. + Tedaj +\begin_inset Formula $G-S$ +\end_inset + + razbije +\begin_inset Formula $G$ +\end_inset + + na dva podgrafa: +\begin_inset Formula $K_{1}\left\{ v_{i+1},\dots,v_{j-1}\right\} $ +\end_inset + + in +\begin_inset Formula $K_{2}=\left(VG\cap\left\{ v_{i},v_{j}\right\} \right)\cap K_{1}=\left\{ v_{j+1},\dots,v_{n},v_{1},\dots,v_{i-1}\right\} $ +\end_inset + +. + Če +\begin_inset Formula $i=j-1$ +\end_inset + +, je ena podgraf prazen, če +\begin_inset Formula $n=3$ +\end_inset + + prav tako. + Podgrafa sta lahko povezana in tvorita skupno komponento, lahko pa nista + in tvorita dve komponenti. + Toda z odstranjevanjem +\begin_inset Formula $\left|S\right|$ +\end_inset + + vozlišč iz cikla lahko napravimo največ +\begin_inset Formula $\left|S\right|$ +\end_inset + + komponent. +\end_layout + +\begin_layout Remark* +Izrek uporabimo v kontrapoziciji: +\begin_inset Formula $\exists S\subseteq VG\ni:\Omega\left(G-S\right)>\left|S\right|\Rightarrow G$ +\end_inset + + ni Hamiltonov. +\end_layout + +\begin_layout Example* +\begin_inset Formula $G$ +\end_inset + + vsebuje prerezno vozlišče +\begin_inset Formula $\Rightarrow G$ +\end_inset + + ni Hamiltonov. +\end_layout + +\begin_layout Claim* +\begin_inset Formula $K_{m,n}$ +\end_inset + + Hamiltonov +\begin_inset Formula $\Leftrightarrow m=n$ +\end_inset + +. +\end_layout + +\begin_layout Proof +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + Uporabimo izrek o potrebnem pogoju. + RAAPDD BSŠ +\begin_inset Formula $m>n$ +\end_inset + +: +\begin_inset Formula $\Omega\left(G-n\right)=m$ +\end_inset + +, kar vodi v +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +. + +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + + Očitno lahko skiciramo Hamiltonov cikel. +\end_layout + +\begin_layout Claim* +\begin_inset Formula $G$ +\end_inset + + dvodelen z razdelitvijo +\begin_inset Formula $\left(A,B\right)$ +\end_inset + +, +\begin_inset Formula $\left|A\right|\not=\left|B\right|\Rightarrow G$ +\end_inset + + ni Hamiltonov. +\end_layout + +\begin_layout Subsection +Navedite Orejev zadostni pogoj za obstoj Hamiltonovega cikla v grafu. + Skicirajte dokaz tega izreka. +\end_layout + +\begin_layout Theorem* +Ore: +\begin_inset Formula $G$ +\end_inset + + graf, +\begin_inset Formula $\left|VG\right|\geq3$ +\end_inset + +, +\begin_inset Formula $\left(\forall u,v\in VG:uv\not\in EG\Rightarrow\deg u+\deg v\geq\left|VG\right|\right)\Rightarrow G$ +\end_inset + + Hamiltonov. + ZDB če za vsak par nesosednjih vozlišč v grafu z vsaj tremi vozlišči velja + +\begin_inset Formula $\deg u+\deg v\geq\left|VG\right|$ +\end_inset + +, je graf Hamiltonov. +\end_layout + +\begin_layout Proof +Dokaz z metodo najmanjšega protiprimera. + RAAPDD izrek ne velja. + Tedaj +\begin_inset Formula $\exists G$ +\end_inset + +, da predpostavka velja, zaključek pa ne. + Med vsemi takimi grafi izberimo tiste z najmanj vozlišči, izmed njih pa + enega izmed tistih, ki imajo največ povezav, in ga fiksiramo. + Naj bo to graf +\begin_inset Formula $G$ +\end_inset + +. + Zanj velja: +\end_layout + +\begin_deeper +\begin_layout Itemize +\begin_inset Formula $\forall u,v\in VG:uv\not\in EG\Rightarrow\deg u+\deg v\geq\left|VG\right|$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $G$ +\end_inset + + ni Hamiltonov +\begin_inset Formula $\Rightarrow G$ +\end_inset + + gotovo ni polni graf +\begin_inset Formula $\Rightarrow\exists u,v\in VG\ni:uv\not\in EG$ +\end_inset + +. + Naj bo +\begin_inset Formula $H$ +\end_inset + + graf, da velja +\begin_inset Formula $VH\coloneqq VG$ +\end_inset + + in +\begin_inset Formula $EH\coloneqq EG\cup uv$ +\end_inset + +. + Zanj še vedno velja prejšnja točka, zaradi izbire +\begin_inset Formula $G$ +\end_inset + + (največ povezav) pa ni več protiprimer za izrek, zato je Hamiltonov. + Vsak Hamiltonov cikel v +\begin_inset Formula $H$ +\end_inset + + vsebuje +\begin_inset Formula $uv$ +\end_inset + +, sicer bi obstajal že v +\begin_inset Formula $G$ +\end_inset + +. + Vseeno pa +\begin_inset Formula $G$ +\end_inset + + premore Hamiltonovo pot, saj smo do cikla namreč dodali le eno povezavo. + Naj bo +\begin_inset Formula $\left[u=v_{1},v_{2},\dots,v_{n-1},v_{n}=v\right]$ +\end_inset + + Hamiltonov cikel v +\begin_inset Formula $H$ +\end_inset + +. + Vpeljimo množici +\begin_inset Formula $S=\left\{ v_{i},uv_{i+1}\in EG\right\} $ +\end_inset + + (ZDB predhodniki sosedov +\begin_inset Formula $u$ +\end_inset + + na Hamiltonovi poti v +\begin_inset Formula $G$ +\end_inset + +) in +\begin_inset Formula $T=\left\{ v_{i},vv_{i}\in EG\right\} $ +\end_inset + + (ZDB sosedje +\begin_inset Formula $v$ +\end_inset + + na Hamiltonovi poti v +\begin_inset Formula $G$ +\end_inset + +). + Velja +\begin_inset Formula $\left|S\cup T\right|=\left|S\right|+\left|T\right|-\left|S\cap T\right|$ +\end_inset + +, torej +\begin_inset Formula $\left|S\cup T\right|+\left|S\cup T\right|=\left|S\right|+\left|T\right|=\deg_{G}u+\deg_{G}v\overset{\text{predpostavka}}{\geq}\left|VG\right|=n$ +\end_inset + +. + Toda ker je +\begin_inset Formula $\left|S\cup T\right|=\left|VG\right|$ +\end_inset + +, +\begin_inset Formula $\left|S\cap T\right|\not=\emptyset$ +\end_inset + +, torej ima +\begin_inset Formula $v$ +\end_inset + + soseda iz +\begin_inset Formula $S$ +\end_inset + + (recimo mu +\begin_inset Formula $v_{i}$ +\end_inset + +), torej lahko konstruiramo Hamiltonov cikel v +\begin_inset Formula $G$ +\end_inset + +: +\begin_inset Formula $\left[u=v_{1},v_{2},\dots,v_{i},v_{n}=v,v_{n-1},\dots,v_{i+1},v_{1}=u\right]$ +\end_inset + + ( +\begin_inset Formula $v_{i+1}$ +\end_inset + + je namreč po konstrukciji +\begin_inset Formula $S$ +\end_inset + + sosed +\begin_inset Formula $u$ +\end_inset + +), kar vodi v +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Subsection +\begin_inset CommandInset label +LatexCommand label +name "subsec:Kaj-so-ravninski" + +\end_inset + +Kaj so ravninski grafi? Kaj so lica ravninske vložitve grafa in čemu je + enaka vsota dolžin vseh lic ravninske vložitve grafa? Kako lahko omejimo + število povezav ravninskega grafa s pomočjo njegove ožine? +\end_layout + +\begin_layout Definition* +Graf +\begin_inset Formula $G$ +\end_inset + + ravninski +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + lahko ga narišemo v ravnino tako, da se nobeni povezavi ne križata. + Ravninski graf skupaj z ustrezno risbo (vložitvijo) je graf, vložen v ravnino. +\end_layout + +\begin_layout Example* +\begin_inset Formula $K_{2,3}$ +\end_inset + + je ravninski, +\begin_inset Formula $K_{3,3}$ +\end_inset + + ni ravninski. +\end_layout + +\begin_layout Theorem* +Jordan: Sklenjena enostavna krivulja (t. + j. + taka, ki same sebe ne križa) v ravnini razdeli ravnino v notranjost, zunanjost + in krivuljo samo. +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $G$ +\end_inset + + ravninski, vložen v ravnino. + Sklenjena območja v ravnini, dobljena tako, da iz risbe odstranimo točke, + ki ustrezajo vozliščem in povezavam, imenujemo +\series bold +lica vložitve +\series default +. + S +\begin_inset Formula $FG$ +\end_inset + + označimo množico lič vložitve. + Seveda je tudi zunanje/neomejeno območje lice. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\left|FQ_{3}\right|=6$ +\end_inset + + +\end_layout + +\begin_layout Remark* +\begin_inset Formula $G$ +\end_inset + + lahko vložimo v ravnino +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + lahko ga vložimo na sfero. +\end_layout + +\begin_layout Definition* +Dolžina lica +\begin_inset Formula $F$ +\end_inset + + +\begin_inset Formula $(\text{\ensuremath{\ell F}}),$ +\end_inset + +je število povezav, ki jih prehodimo, ko obhodimo lice\SpecialChar endofsentence + +\end_layout + +\begin_layout Remark* +Vsako drevo je ravninski graf. + Ima eno lice, katerega dolžina je +\begin_inset Formula $2\left|ET\right|$ +\end_inset + +. +\end_layout + +\begin_layout Definition* +Ožina grafa +\begin_inset Formula $G$ +\end_inset + +, +\begin_inset Formula $gG$ +\end_inset + +, je dolžina najkrajšega cikla v +\begin_inset Formula $G$ +\end_inset + +. + Če je +\begin_inset Formula $G$ +\end_inset + + gozd, je +\begin_inset Formula $gG\coloneqq\infty$ +\end_inset + +. +\end_layout + +\begin_layout Claim* +Če je +\begin_inset Formula $G$ +\end_inset + + ravninsli graf, vložen v ravnino, velja +\begin_inset Formula +\[ +\sum_{F\in FG}\ell F=2\left|EG\right| +\] + +\end_inset + + +\end_layout + +\begin_layout Claim* +Naj +\begin_inset Formula $G$ +\end_inset + + premore cikel. + Naj bo +\begin_inset Formula $F\in FG$ +\end_inset + +. + +\begin_inset Formula $\ell F\geq gG$ +\end_inset + +. + +\begin_inset Formula $2\left|EG\right|=\sum_{F\in FG}\ell F\geq\sum_{F\in FG}gG=gG\left|FG\right|$ +\end_inset + + +\end_layout + +\begin_layout Claim* +Posledično: Če je +\begin_inset Formula $G$ +\end_inset + + ravninski graf z vsaj enim ciklom in je vložen v ravnino, je +\begin_inset Formula $\left|EG\right|\geq\frac{gG}{2}\left|FG\right|$ +\end_inset + +(*). +\end_layout + +\begin_layout Subsection +Kaj pravi Eulerjeva formula za ravninske grafe? Skicirajte njen dokaz. + Katere posledice Eulerjeve formule poznate? +\end_layout + +\begin_layout Theorem* +Eulerjeva formula: Če je +\begin_inset Formula $G$ +\end_inset + + ravninski vložen v ravnino, velja +\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=1+\Omega G$ +\end_inset + +. +\end_layout + +\begin_layout Example* +Graf +\begin_inset Quotes gld +\end_inset + +tri hiše +\begin_inset Quotes grd +\end_inset + +: +\begin_inset Formula $\left|VG\right|=15$ +\end_inset + +, +\begin_inset Formula $\left|EG\right|=18$ +\end_inset + +, +\begin_inset Formula $\left|FG\right|=7$ +\end_inset + +, +\begin_inset Formula $\Omega G=3$ +\end_inset + +, +\begin_inset Formula $15+16+7=1+3$ +\end_inset + + +\end_layout + +\begin_layout Proof +Dokažimo najprej za povezan multigraf +\begin_inset Formula $G$ +\end_inset + +. + Dokazujemo +\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=2$ +\end_inset + +. + Indukcija po številu vozlišč: +\end_layout + +\begin_layout Proof +Baza: Izolirano vozlišče: +\begin_inset Formula $\left|VG\right|=1$ +\end_inset + +, +\begin_inset Formula $\left|EG\right|=0+z$ +\end_inset + +, +\begin_inset Formula $\left|FG\right|=1+z$ +\end_inset + +, kjer je +\begin_inset Formula $z$ +\end_inset + + število zank (za navaden graf +\begin_inset Formula $z=0$ +\end_inset + +). + Drži. +\end_layout + +\begin_layout Proof +Korak: Naj bo +\begin_inset Formula $e\in EG$ +\end_inset + + poljubna. + Skrčimo jo (kot v multigrafu). + +\begin_inset Formula $\left|V\left(G/e\right)\right|=\left|VG\right|-1$ +\end_inset + +, +\begin_inset Formula $\left|E\left(G/e\right)\right|=\left|EG\right|-1$ +\end_inset + +, +\begin_inset Formula $\left|F\left(G/e\right)\right|=\left|FE\right|$ +\end_inset + +. + Velja +\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=2$ +\end_inset + +, saj po I. + P. + velja +\begin_inset Formula $\left|V\left(G/e\right)\right|+1-\left|E\left(G/e\right)\right|-1+\left|F\left(G/e\right)\right|=2$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Sedaj dokažimo še za nepovezan multigraf +\begin_inset Formula $G$ +\end_inset + + z +\begin_inset Formula $\Omega G$ +\end_inset + + komponentami. + Grafu lahko dodamo +\begin_inset Formula $\Omega G-1$ +\end_inset + + povezav, da ga povežemo, s čimer ne spremenimo niti +\begin_inset Formula $\left|FG\right|$ +\end_inset + + niti +\begin_inset Formula $\left|VG\right|$ +\end_inset + +. + Če je +\begin_inset Formula $E$ +\end_inset + + množica povezav, ki jo moramo dodati, velja +\begin_inset Formula $\left|VG\right|-\left|EG\cup E\right|+\left|FG\right|=2=\left|VG\right|-\left|EG\right|-\Omega G+1+\left|FG\right|\Rightarrow\left|VG\right|-\left|EG\right|+\left|FG\right|=2-1+\Omega G=1+\Omega G$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Za ravninski graf z vsaj tremi vozlišči velja +\begin_inset Formula $\left|EG\right|\leq3\left|VG\right|-6$ +\end_inset + +, če je slednji brez trikotnikov, a ima cikel, pa celo +\begin_inset Formula $\left|EG\right|\leq2\left|VG\right|-4$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Dokažimo za povezan ravninski graf, sicer mu lahko samo dodamo povezave + in ga povežemo. + Ločimo primera: +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(G\text{ drevo}\right)$ +\end_inset + + +\begin_inset Formula $\left|EG\right|=\left|VG\right|-1$ +\end_inset + + (karakterizacija dreves) +\begin_inset Formula $\overset{?}{\leq}3\left|VG\right|-6$ +\end_inset + +. + Drži, kajti +\begin_inset Formula $\left|VG\right|\geq3$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(G\text{ premore cikel}\right)$ +\end_inset + + Po Eulerjevi formuli velja +\begin_inset Formula +\[ +2=\left|VG\right|-\left|EG\right|+\left|FG\right|\overset{\text{(*) iz \ref{subsec:Kaj-so-ravninski}}}{\leq}\left|VG\right|-\left|EG\right|+\frac{2\left|EG\right|}{gG} +\] + +\end_inset + + +\begin_inset Formula +\[ +2\leq\left|VG\right|-\left|EG\right|+\frac{2\left|EG\right|}{gG} +\] + +\end_inset + + +\begin_inset Formula +\[ +2-\left|VG\right|\leq\left|EG\right|\left(\frac{2}{gG}-1\right)\quad\quad\quad\quad/^{-1} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left|VG\right|-2\geq\left|EG\right|\left(1-\frac{2}{gG}\right)=\left|EG\right|\left(\frac{gG-2}{gG}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\left(\left|VG\right|-2\right)\frac{gG}{gG-2}\geq\left|EG\right| +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\frac{gG}{gG-2}$ +\end_inset + + je največ 3 in to tedaj, ko graf premore trikotnik. + Če za vse večje +\begin_inset Formula $gG$ +\end_inset + + bo ta ulomek manjši, torej lahko levo stran omejimo navzdol: +\begin_inset Formula $3\left|VG\right|-6\geq\left(\left|VG\right|-2\right)\frac{gG}{gG-2}\geq\left|EG\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +Če pa graf ne premore trikotnika, a ima cikel, pa je +\begin_inset Formula $\frac{gG}{gG-2}=2$ +\end_inset + + in levo stran strožje omejimo navzgor in velja +\begin_inset Formula $2\left|VG\right|-4\geq\left|EG\right|$ +\end_inset + +. + +\end_layout + +\end_deeper +\begin_layout Subsection +Kaj je kromatično število +\begin_inset Formula $\chi\left(G\right)$ +\end_inset + + grafa +\begin_inset Formula $G$ +\end_inset + +? Pojasnite požrešni algoritem barvanja grafa. + Kako lahko z njegovo pomočjo navzgor omejimo +\begin_inset Formula $\chi\left(G\right)$ +\end_inset + +? +\end_layout + +\begin_layout Definition* +\begin_inset Formula $k-$ +\end_inset + +barvanje grafa +\begin_inset Formula $G$ +\end_inset + + je preslikava +\begin_inset Formula $C:VG\to\left\{ 1..k\right\} $ +\end_inset + +, za katero velja +\begin_inset Formula $uv\in EG\Rightarrow Cu\not=Cv$ +\end_inset + +. + Kromatično število +\begin_inset Formula $\chi G$ +\end_inset + + je najmanjši +\begin_inset Formula $k$ +\end_inset + +, za katerega najdemo +\begin_inset Formula $k-$ +\end_inset + +barvanje +\begin_inset Formula $G$ +\end_inset + +. + Za fiksen +\begin_inset Formula $i$ +\end_inset + + je +\begin_inset Formula $\left\{ u\in VG;Cu=i\right\} $ +\end_inset + + barvni razred, ki je neodvisna množica +\begin_inset Foot +status open + +\begin_layout Plain Layout +Za neodvisno množico +\begin_inset Formula $S\subseteq VG$ +\end_inset + + velja +\begin_inset Formula $\forall u,v\in S:uv\not\in EG$ +\end_inset + + ZDB je +\begin_inset Quotes gld +\end_inset + +brez povezav +\begin_inset Quotes grd +\end_inset + +. + Glej vprašanje +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:Kaj-je-neodvisnostno" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\chi K_{n}=n$ +\end_inset + +, +\begin_inset Formula $\chi C_{n}=\begin{cases} +2 & ;n\text{ sod}\\ +3 & ;n\text{ lih} +\end{cases}$ +\end_inset + +, +\begin_inset Formula $\chi P_{5,2}=3$ +\end_inset + +. +\end_layout + +\begin_layout Definition* +Klično število grafa +\begin_inset Formula $G$ +\end_inset + + označimo z +\begin_inset Formula $\omega G$ +\end_inset + +. + Velja +\begin_inset Formula $\omega G=\left|VH\right|$ +\end_inset + +, kjer je +\begin_inset Formula $H$ +\end_inset + + največji poln podgraf v +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Remark* +\begin_inset Formula $\forall G\in\mathcal{G}:\chi G\geq\omega G$ +\end_inset + +. +\end_layout + +\begin_layout Definition* +Požrešni algoritem barvanja: V poljubnem vrstnem redu zaporedno barvamo + vozlišča. + Posameznemu vozlišču priredimo najnižjo barvo, ki še ni uporabljena na + njegovih sosedih. +\end_layout + +\begin_layout Fact* +Vedno +\begin_inset Formula $\exists$ +\end_inset + + vrstni red barvanja, da požrešni algoritem vrne barvanje s +\begin_inset Formula $\chi G$ +\end_inset + + barvami. +\end_layout + +\begin_layout Claim* +\begin_inset Formula $\forall G\in\mathcal{G}:\chi G\leq\Delta G+1$ +\end_inset + + ZDB +\begin_inset Formula $\chi G$ +\end_inset + + je kvečjemu 1 večji od največje stopnje v grafu. +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $x_{1},\dots,x_{n}$ +\end_inset + + poljuben vrstni red vozlišč. + Poženemo požrešni algoritem. + Na poljubnem +\begin_inset Formula $i$ +\end_inset + +tem koraku, ko barvamo +\begin_inset Formula $x_{i}$ +\end_inset + +, je kvečjemu +\begin_inset Formula $\deg_{G}x_{i}$ +\end_inset + + barv, ki niso na razpolago, kar je +\begin_inset Formula $\leq\Delta G$ +\end_inset + +. +\end_layout + +\begin_layout Claim* +\begin_inset Formula $G$ +\end_inset + + ni regularen +\begin_inset Formula $\Rightarrow\forall G\in\mathcal{G}:\chi G\leq\Delta G$ +\end_inset + + +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $x$ +\end_inset + + tisto vozlišče, ki največje stopnje (*). + Vzamemo ga kot koren za BFS in z BFS vozlišča zmečemo v zaporedje +\begin_inset Formula $a$ +\end_inset + +. + Nato požrešno barvamo v obratni smeri, kot jo določa +\begin_inset Formula $a$ +\end_inset + +. + Na vsakem koraku, razen na korenu, bomo imeli soseda, ki še ni pobarvan, + torej je kvečjemu +\begin_inset Formula $\deg_{G}x_{i}-1$ +\end_inset + + barv, ki niso na razpolago, kar je +\begin_inset Formula $\leq\Delta G$ +\end_inset + +. + Na zadnjem koraku (koren) pa po predpostavki (*) na razpolago ni kvečjemu + +\begin_inset Formula $\Delta G-1$ +\end_inset + + barv. +\end_layout + +\begin_layout Theorem* +Brooks: +\begin_inset Formula $G$ +\end_inset + + povezan, +\begin_inset Formula $G$ +\end_inset + + niti poln niti lihi cikel +\begin_inset Formula $\Rightarrow\chi G\leq\Delta G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Theorem* +Naj bo +\begin_inset Formula $d_{1}\geq d_{2}\geq\cdots\geq d_{n}$ +\end_inset + + zaporedje stopenj grafa +\begin_inset Formula $G$ +\end_inset + +. + Tedaj velja +\begin_inset Formula $\chi G\leq1+\max_{i=1}^{n}\left(\min\left\{ d_{i},i-1\right\} \right)$ +\end_inset + + +\end_layout + +\begin_layout Proof +Poženimo požrešni algoritem barvanja v padajočem zaporedju stopenj. + Na +\begin_inset Formula $i$ +\end_inset + +tem barvamo vozlišče stopnje +\begin_inset Formula $d_{i}$ +\end_inset + +, zato imamo gotovo na voljo barvo iz +\begin_inset Formula $\left\{ 1..d_{i}+1\right\} $ +\end_inset + +. + Ker smo doslej pobarvali zgolj +\begin_inset Formula $i-1$ +\end_inset + + vozlišč, imamo gotovo na voljo barvo iz +\begin_inset Formula $\left\{ 1..i\right\} $ +\end_inset + +. + Algoritem pobarva vozlišče z barvo +\begin_inset Formula $\leq\min\left\{ i,d_{i}+1\right\} $ +\end_inset + +. + Največja uporabljena barva +\begin_inset Formula $k$ +\end_inset + + je +\begin_inset Formula $\leq\max_{i=1}^{n}\left\{ \min\left\{ d_{i}+1,i\right\} \right\} $ +\end_inset + +. + Torej +\begin_inset Formula $\chi G\leq1+\max_{i=1}^{n}\left(\min\left\{ d_{i},i-1\right\} \right)$ +\end_inset + + (izven +\begin_inset Formula $\max$ +\end_inset + + smo prišteli 1, znotraj +\begin_inset Formula $\min$ +\end_inset + + smo od vseh elementov 1 odšteli). +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO XXX FIXME dokaz in trditev za barvanje ravninskega grafa s petimi barvami +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Kaj je kromatični indeks +\begin_inset Formula $\chi'\left(G\right)$ +\end_inset + + grafa +\begin_inset Formula $G$ +\end_inset + +? Kaj pravi Vizingov izrek in kako na njegovi osnovi razdelimo vse grafe + v dva razreda? +\end_layout + +\begin_layout Definition* +\begin_inset Formula $k-$ +\end_inset + +barvanje povezav je preslikava +\begin_inset Formula $C:EG\to\left\{ 1..k\right\} \ni:uv,uw\in EG\Rightarrow C\left(uv\right)\not=C\left(uw\right)$ +\end_inset + +. + ZDB povezavi s skupnim krajiščem dobita različni barvi. + Kromatični indeks grafa +\begin_inset Formula $G$ +\end_inset + + (oznaka +\begin_inset Formula $\chi'G$ +\end_inset + +) je najmanjši +\begin_inset Formula $k$ +\end_inset + +, za katerega +\begin_inset Formula $\exists k-$ +\end_inset + +barvanje grafa +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\chi'C_{n}=\begin{cases} +2 & ;n\text{ sod}\\ +3 & ;n\text{ lih} +\end{cases}$ +\end_inset + + +\end_layout + +\begin_layout Theorem* +Vizing: +\begin_inset Formula $\forall G\in\mathcal{G}:\Delta G\leq\chi'G\leq\Delta G+1$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Prvi neenačaj je očiten, drugega ne bomo dokazali. +\end_layout + +\begin_layout Definition* +Graf +\begin_inset Formula $G$ +\end_inset + + je razreda I, če je +\begin_inset Formula $\chi'G=\Delta G$ +\end_inset + + oziroma razreda II, če je +\begin_inset Formula $\chi'G=\Delta G+1$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $C_{2n}$ +\end_inset + + so razreda +\begin_inset Formula $I$ +\end_inset + +, +\begin_inset Formula $C_{2n+1}$ +\end_inset + + so razreda II, +\begin_inset Formula $K_{3}$ +\end_inset + + je razreda II, +\begin_inset Formula $K_{4}$ +\end_inset + + je razreda +\begin_inset Formula $I$ +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO XXX FIXME dokaz, da so dvodelni grafi I, K_2k razreda I in K_2k+1 razreda + II. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +\begin_inset CommandInset label +LatexCommand label +name "subsec:Kaj-je-neodvisnostno" + +\end_inset + +Kaj je neodvisnostno število grafa? Zanj podajte spodnjo mejo in zgornjo + mejo. + Opišite algoritem za izračun neodvisnostnega števila drevesa. +\end_layout + +\begin_layout Definition* +Če je +\begin_inset Formula $G$ +\end_inset + + graf in +\begin_inset Formula $I\subseteq VG$ +\end_inset + +, je +\begin_inset Formula $I$ +\end_inset + + neodvisna +\begin_inset Formula $\Leftrightarrow\forall u,v\in I:uv\not\in EG$ +\end_inset + + ZDB če nobeni dve vozlišči v I nista sosednji v +\begin_inset Formula $G$ +\end_inset + +. + Moč največje neodvisne množice v +\begin_inset Formula $G$ +\end_inset + + je neodvisnostno število +\begin_inset Formula $G$ +\end_inset + +, označeno z +\begin_inset Formula $\alpha G$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\alpha K_{n}=1$ +\end_inset + +, +\begin_inset Formula $\alpha C_{n}=\lfloor\frac{n}{2}\rfloor$ +\end_inset + +, +\begin_inset Formula $\alpha P_{5,2}=4$ +\end_inset + + +\end_layout + +\begin_layout Claim* +\begin_inset Formula $\forall G\in\mathcal{G}:\alpha G\cdot\chi G\geq\left|VG\right|$ +\end_inset + + +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $\chi G=k$ +\end_inset + + in +\begin_inset Formula $V_{1},\dots,V_{k}$ +\end_inset + + barvni razredi nekega fiksnega barvanja s k barvami. + Slednji so neodvisne množice. + +\begin_inset Formula $\forall i\in\left\{ 1..k\right\} :\left|V_{i}\right|\leq\alpha G\Rightarrow\left|VG\right|=\sum_{i=1}^{k}\left|V_{i}\right|\leq\sum_{i=1}^{k}\alpha G=\alpha G\cdot k=\alpha G\chi G$ +\end_inset + + +\end_layout + +\begin_layout Corollary* +Spodnja meja za neodvisnostno število +\begin_inset Formula $\alpha G\geq\frac{\left|VG\right|}{\chi G}$ +\end_inset + +. +\end_layout + +\begin_layout Claim* +Zgornja meja za neodvisnostno število: +\begin_inset Formula $\alpha G\leq\left|VG\right|-\frac{\left|EG\right|}{\Delta G}$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $I$ +\end_inset + + poljubna največja neodvisna množica v +\begin_inset Formula $G$ +\end_inset + +, torej +\begin_inset Formula $\left|I\right|=\alpha G$ +\end_inset + +. + V +\begin_inset Formula $VG\setminus I$ +\end_inset + + je vozlišč +\begin_inset Formula $\left|VG\right|-\alpha G$ +\end_inset + +. + Ker vozlišča v +\begin_inset Formula $I$ +\end_inset + + medsebojno niso povezana, +\begin_inset Formula $\left|EG\right|\leq\left(\left|VG\right|-\alpha G\right)\cdot\Delta G\Rightarrow\left|EG\right|\leq\left|VG\right|\Delta G-\alpha G\Delta G\Rightarrow\alpha G\Delta G\leq\left|VG\right|\Delta G-\left|EG\right|\Rightarrow\alpha G\leq\left|VG\right|-\frac{\left|EG\right|}{\Delta G}$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $Q_{d},d\geq1$ +\end_inset + +: Zgornja meja: +\begin_inset Formula $\alpha Q_{d}\leq\left|VQ_{d}\right|-\frac{\left|EQ_{d}\right|}{\Delta Q_{d}}=2^{d}-\frac{d2^{d-1}}{d}=2^{d}-2^{d-1}=2^{d-1}$ +\end_inset + +, Spodnja meja: +\begin_inset Formula $\alpha Q_{d}\geq\frac{\left|VQ_{d}\right|}{\chi Q_{d}}=\frac{2^{d}}{2}=2^{d-1}$ +\end_inset + +, torej +\begin_inset Formula $\alpha Q_{d}=2^{d-1}$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph* +Neodvisnostno število dreves +\end_layout + +\begin_layout Standard +Naj bo +\begin_inset Formula $T$ +\end_inset + + drevo s poljubnim korenom +\begin_inset Formula $r\in VT$ +\end_inset + +. + Odslej na drevo glejmo +\begin_inset Formula $T$ +\end_inset + + kot na drevo s korenom +\begin_inset Formula $r$ +\end_inset + +. + Za neodvisno množico +\begin_inset Formula $S$ +\end_inset + + drevesa +\begin_inset Formula $T$ +\end_inset + + in poljubno vozlišče +\begin_inset Formula $x\in VT$ +\end_inset + + velja: +\begin_inset Formula $x\in S\Rightarrow$ +\end_inset + + potomci (otroci) +\begin_inset Formula $x\not\in S$ +\end_inset + +. + Če pa +\begin_inset Formula $x\not\in S$ +\end_inset + +, pa +\begin_inset Formula $S$ +\end_inset + + sme vsebovati potomce +\begin_inset Formula $x$ +\end_inset + +. + Z +\begin_inset Formula $Iv$ +\end_inset + + označimo velikost največje neodvisne množice s korenom v +\begin_inset Formula $v\in VT$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Očitno velja +\begin_inset Formula $\alpha T=Ir$ +\end_inset + +. + Z rekurzivnim postopkom določimo +\begin_inset Formula $\text{\ensuremath{\alpha T}}$ +\end_inset + + na tak način: +\begin_inset Formula +\[ +\alpha T=Ir=\text{\ensuremath{\max}\left\{ 1+\sum_{v\in\text{vnuki/drugi potomci \ensuremath{r}}}Iv,\sum_{v\in\text{otroci/potomci }r}Iv\right\} } +\] + +\end_inset + +Formula je očitna. + Na vsakem rekurzivnem koraku lahko bodisi v množico izberemo +\begin_inset Formula $v$ +\end_inset + + in ne izberemo njegovih potomcev, bodisi izberemo potomce, njega pa ne. + Z rekurzivnim algoritmom preverimo vse možnosti. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\alpha T_{n}$ +\end_inset + +, kjer je +\begin_inset Formula $T_{n}$ +\end_inset + + polno dvojiško drevo globine +\begin_inset Formula $n\in\mathbb{N}_{0}$ +\end_inset + +: Vpeljimo zaporedje +\begin_inset Formula $a_{n}=\alpha T_{n}$ +\end_inset + + in velja: +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}_{0}}=\left[1,2,5,10,21,42,\dots\right]$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +\begin_inset Formula $a_{0}=1$ +\end_inset + +, +\begin_inset Formula $a_{1}=2$ +\end_inset + +, za +\begin_inset Formula $n\geq2$ +\end_inset + +: +\begin_inset Formula $a_{n}=\begin{cases} +2_{n-1}+1 & ;n\text{ sod}\\ +2_{n-1} & ;n\text{ lih} +\end{cases}$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Z indukcijo. + Baza sta +\begin_inset Formula $a_{0}$ +\end_inset + + in +\begin_inset Formula $a_{1}$ +\end_inset + +. + Indukcijska predpostavka je dana v izreku. + ITD DOPIŠI DOKAZ. + +\begin_inset Quotes gld +\end_inset + +DS2P FMF 2024-04-18 +\begin_inset Quotes grd +\end_inset + + stran 5. +\end_layout + +\begin_layout Subsection +\begin_inset CommandInset label +LatexCommand label +name "subsec:Kaj-je-dominacijsko" + +\end_inset + +Kaj je dominacijsko število grafa? Zanj podajte spodnjo mejo in zgornjo + mejo. + Kakšna je zveza med dominacijskim številom grafa in njegovega vpetega podgrafa? +\end_layout + +\begin_layout Standard +Naj bo +\begin_inset Formula $G$ +\end_inset + + graf. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Neodvisna množica +\begin_inset Formula $G$ +\end_inset + + je maksimalna, če ni prava podmnožica kakšne neodvisne množice v +\begin_inset Formula $G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +\begin_inset Formula $N_{G}\left(u\right)\coloneqq\left\{ v;uv\in EG\right\} $ +\end_inset + + je soseščina vozlišča +\begin_inset Formula $u$ +\end_inset + + (sosedje +\begin_inset Formula $u$ +\end_inset + + v +\begin_inset Formula $G$ +\end_inset + +). + +\begin_inset Formula $N_{G}\left[u\right]\coloneqq N_{G}\left(u\right)\cup\left\{ u\right\} $ +\end_inset + + je zaprta soseščina vozlišča +\begin_inset Formula $u$ +\end_inset + +. + +\begin_inset Formula $N_{G}\left[D\right]=\bigcup_{u\in D}N_{G}\left[u\right]$ +\end_inset + + je zaprta soseščina množice vozlišč +\begin_inset Formula $D.$ +\end_inset + + +\begin_inset Formula $D\subseteq VG$ +\end_inset + + dominira +\begin_inset Formula $X\subseteq VG\Leftrightarrow X\subseteq N_{G}\left[D\right]$ +\end_inset + +. + Če +\begin_inset Formula $D$ +\end_inset + + dominira +\begin_inset Formula $VG$ +\end_inset + +, pravimo, da je +\begin_inset Formula $D$ +\end_inset + + dominantna množica grafa +\begin_inset Formula $G$ +\end_inset + +. + ZDB +\begin_inset Formula $D$ +\end_inset + + dominantna množica +\begin_inset Formula $G\Leftrightarrow\forall u\in VG:u\in D\vee\exists x\in D\ni:xu\in EG$ +\end_inset + + ZDB vsako vozlišče je v +\begin_inset Formula $D$ +\end_inset + + ali pa ima soseda iz +\begin_inset Formula $D$ +\end_inset + +. + Moč najmanjše dominacijske množice za +\begin_inset Formula $G$ +\end_inset + + je dominacijsko število grafa +\begin_inset Formula $G$ +\end_inset + +, označeno z +\begin_inset Formula $\gamma G$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Remark* +Vsaka maksimalna neodvisna množica grafa je njegova dominantna množica. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\gamma K_{n}=1$ +\end_inset + +, +\begin_inset Formula $\gamma C_{n}=\lceil\frac{n}{3}\rceil$ +\end_inset + +, +\begin_inset Formula $\gamma P_{5,2}=3$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Za vsak graf brez izoliranih vozlišč velja, da je +\begin_inset Formula $\lceil\frac{\left|VG\right|}{\Delta G+1}\rceil\leq\gamma G\leq\lfloor\frac{\left|VG\right|}{2}\rfloor$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Spodnja meja: Če je +\begin_inset Formula $G$ +\end_inset + + dominantna množica in +\begin_inset Formula $u\in D$ +\end_inset + +, potem +\begin_inset Formula $u$ +\end_inset + + dominira +\begin_inset Formula $\leq\deg_{G}u+1$ +\end_inset + + vozlišč, torej vsako vozlišče iz +\begin_inset Formula $D$ +\end_inset + + dominira kvečjemu +\begin_inset Formula $\Delta G+1$ +\end_inset + + vozlišč. + Ker +\begin_inset Formula $VG\subseteq N_{G}\left[D\right]$ +\end_inset + + (unija zaprtih okolic vozlišč iz dominantne množice prekrije cel graf), + je +\begin_inset Formula $\left|D\right|\geq\frac{\left|VG\right|}{\Delta G+1}$ +\end_inset + +, kajti +\begin_inset Formula $\left|VG\right|=\left|N_{G}\left[D\right]\right|=\left|\bigcup_{u\in D}N_{G}\left[u\right]\right|\leq\sum_{u\in D}N\left(u\right)\leq\sum_{u\in D}$ +\end_inset + + +\begin_inset Formula $\left(\Delta G+1\right)=\gamma G\left(\Delta G+1\right)$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Zgornja meja: Naj bo +\begin_inset Formula $I$ +\end_inset + + poljubna maksimalna neodvisna množica. + Vemo, da je +\begin_inset Formula $I$ +\end_inset + + tedaj dominantna. + Če je +\begin_inset Formula $\left|I\right|\leq\frac{\left|VG\right|}{2}$ +\end_inset + + smo dokazali, sicer vzemimo njen komplement +\begin_inset Formula $I'\coloneqq VG\setminus I$ +\end_inset + +. + Trdimo, da je +\begin_inset Formula $I'$ +\end_inset + + dominantna. + Vzemimo poljubno +\begin_inset Formula $u\in G$ +\end_inset + +. + Če +\begin_inset Formula $u\in I'$ +\end_inset + +, dominira samega sebe, sicer, ker je +\begin_inset Formula $G$ +\end_inset + + brez izoliranih vozlišč, ima +\begin_inset Formula $u$ +\end_inset + + vsaj enega soseda, ker pa je +\begin_inset Formula $I$ +\end_inset + + neodvisna, je ta sosed v +\begin_inset Formula $I'$ +\end_inset + +, torej je +\begin_inset Formula $I'$ +\end_inset + + dominantna in ima +\begin_inset Formula $\leq\frac{\left|VG\right|}{2}$ +\end_inset + + vozlišč in velja +\begin_inset Formula $\gamma G\leq\min\left\{ \left|I\right|,\left|I'\right|\right\} \leq\frac{\left|VG\right|}{2}$ +\end_inset + +, kajti +\begin_inset Formula $I\cup I'=VG$ +\end_inset + +. +\end_layout + +\begin_layout Example* +Enakost spodnje meje velja v +\begin_inset Formula $K_{n}$ +\end_inset + +, +\begin_inset Formula $C_{n}$ +\end_inset + +. + Enakost zgornje meje velja pri glavnikih +\begin_inset Formula $T_{n}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +dodaj 2-pakiranje in 2-pakirno število +\begin_inset Formula $\rho$ +\end_inset + +! +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Theorem* +Če je +\begin_inset Formula $H$ +\end_inset + + vpet podgraf +\begin_inset Formula $G$ +\end_inset + +, je +\begin_inset Formula $\gamma G\leq\gamma H$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Naj bo +\begin_inset Formula $D$ +\end_inset + + minimalna dominantna množica za +\begin_inset Formula $H$ +\end_inset + +. + +\begin_inset Formula $\left|D\right|=\gamma H$ +\end_inset + +. + Tedaj je +\begin_inset Formula $D$ +\end_inset + + očitno tudi dominantna množica za +\begin_inset Formula $G$ +\end_inset + +. + Toda seveda se lahko zgodi, da je v +\begin_inset Formula $G$ +\end_inset + + moč najti manjšo dominantno množico kot v +\begin_inset Formula $H$ +\end_inset + +, ker ima +\begin_inset Formula $G$ +\end_inset + + lahko dodatne povezave. +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO XXX FIXME vpeto drevo in +\begin_inset Formula $\gamma$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Kaj je dominacijsko število grafa in kaj je celotno dominacijsko število + grafa? Kakšna je zveza med njima? Kakšna je povezava med dominacijskim + številom grafa in kromatičnim številom komplementa? +\end_layout + +\begin_layout Standard +Dominacijsko število grafa je definirano v vprašanju +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:Kaj-je-dominacijsko" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Definition* +Dominantna množica +\begin_inset Formula $D$ +\end_inset + + grafa +\begin_inset Formula $G$ +\end_inset + + je povezana, če inducira povezan podgraf, neodvisna, če inducira podgraf + brez povezav in +\series bold +celotna +\series default +, če ima vsako vozlišče iz +\begin_inset Formula $VG$ +\end_inset + + soseda v +\begin_inset Formula $D$ +\end_inset + + (tudi vozlišča iz +\begin_inset Formula $D$ +\end_inset + +). + Velikost najmanjše povezane dominantne množice označimo z +\begin_inset Formula $\gamma_{c}G$ +\end_inset + +, neodvisne z +\begin_inset Formula $\gamma_{i}G$ +\end_inset + + in +\series bold +celotne +\series default + z +\begin_inset Formula $\gamma_{t}G$ +\end_inset + + (connected, independent in total). +\end_layout + +\begin_layout Example* +\begin_inset Formula $\gamma_{t}K_{n}=2$ +\end_inset + +, +\begin_inset Formula $\gamma_{t}P_{n}=\lfloor\frac{n}{2}\rfloor+\lceil\frac{n}{2}\rceil-\lfloor\frac{n}{4}\rfloor$ +\end_inset + + (gledamo parčke, ki se dominirajo). +\end_layout + +\begin_layout Theorem* +Za vsak graf brez izoliranih vozlišč velja +\begin_inset Formula $\gamma G\leq\gamma_{t}G\leq2\gamma G$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Spodnja meja je očitna, kajti biti celotna dominantna množica je strožji + pogoj kot biti dominantna množica. + Dokažimo +\begin_inset Formula $\gamma_{t}G\leq2\gamma G$ +\end_inset + +: Naj bo +\begin_inset Formula $D$ +\end_inset + + poljubna dominantna množica +\begin_inset Formula $G$ +\end_inset + +. + Vsakemu +\begin_inset Formula $x\in D$ +\end_inset + + priredimo nekega soseda od +\begin_inset Formula $x$ +\end_inset + +, recimo +\begin_inset Formula $x'$ +\end_inset + + in ga dodajmo v +\begin_inset Formula $\hat{D}$ +\end_inset + +, torej +\begin_inset Formula $\hat{D}=D\cup\left\{ x';x\in D\right\} $ +\end_inset + +, s čimer dominantno množico spremenimo v celotno tako, da ji kvečjemu podvojimo + število vozlišč. +\end_layout + +\begin_layout Example* +Enakost spodnje meje se pojavi pri +\begin_inset Formula $\gamma C_{4}=\gamma_{t}C_{4}=2$ +\end_inset + +, neenakost pa pri recimo +\begin_inset Formula $\gamma G_{3}=2\not=4=\gamma_{t}Q_{3}$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Kaj je grupoid, polgrupa, monoid, grupa? Kako v monoidu izračunamo inverz + produkta obrnljivih elementov? Kako definiramo potence v monoidu in kateri + računski zakoni veljajo zanje? +\end_layout + +\begin_layout Definition* +Če uvedemo binarno operacijo +\begin_inset Formula $f$ +\end_inset + + na množici +\begin_inset Formula $A$ +\end_inset + + takole: +\begin_inset Formula $f:A\times A\to A,f$ +\end_inset + + preslikava, je par +\begin_inset Formula $\left(A,f\right)$ +\end_inset + + +\series bold +grupoid +\series default +. + ZDB zahtevamo, da je operacija preslikava (domena je enaka definicijskemu + območju) in s tem zaprtost operacije. + Dvojiški operator +\begin_inset Formula $f\left(a,b\right)$ +\end_inset + + pišimo kot +\begin_inset Formula $a\cdot b$ +\end_inset + +. + +\end_layout + +\begin_layout Definition* +Če +\begin_inset Formula $\forall a,b,c\in A:\left(a\cdot b\right)\cdot c=a\cdot\left(b\cdot c\right)$ +\end_inset + + (asociativnost), pravimo, da je +\begin_inset Formula $\left(A,\cdot\right)$ +\end_inset + + +\series bold +podgrupa +\series default + ZDB asocietiven grupoid. +\end_layout + +\begin_layout Definition* +Enota je element +\begin_inset Formula $e\in A\ni:\forall a\in A:a\cdot e=e\cdot a=a$ +\end_inset + +. + Polgrupi z enoto pravimo +\series bold +monoid +\series default +. + +\end_layout + +\begin_layout Definition* +Monoidu, v katerem so vsi elementi obrnljivi, pravimo +\series bold +grupa +\series default + ZDB +\series bold + +\begin_inset Formula $\forall a\in A\exists b\in A\ni:a\cdot b=b\cdot a=e$ +\end_inset + + +\series default +. +\end_layout + +\begin_layout Remark* +Ker so inverzi v monoidu enolični (dokaz pri LA), inverz +\begin_inset Formula $a$ +\end_inset + + označimo z +\begin_inset Formula $a^{-1}$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\left(\mathcal{G},\square\right)$ +\end_inset + + monoid, +\begin_inset Formula $\left(\mathbb{R}_{0}^{+}=\left\{ x\in\mathbb{R};x\geq0\right\} ,\max\right)$ +\end_inset + + monoid, množica vseh nizov/seznamov s concat operacijo je monoid. +\end_layout + +\begin_layout Theorem* +Če sta +\begin_inset Formula $a$ +\end_inset + + in +\begin_inset Formula $b$ +\end_inset + + v monoidu obrnljiva, je obrnljiv tudi njun produkt in velja +\begin_inset Formula $\left(a\cdot b\right)^{-1}=b^{-1}\cdot a^{-1}$ +\end_inset + +. +\end_layout + +\begin_layout Proof +\begin_inset Formula $\left(a\cdot b\right)\cdot\left(b^{-1}\cdot a^{-1}\right)=a\cdot\left(b\cdot b^{-1}\right)\cdot a^{-1}=a\cdot e\cdot a^{-1}=a\cdot a^{-1}=e$ +\end_inset + + in podobno +\begin_inset Formula $\left(b^{-1}a^{-1}\right)\left(ab\right)=e$ +\end_inset + +, torej +\begin_inset Formula $\left(b^{-1}a^{-1}\right)\left(ab\right)=\left(ab\right)\left(b^{-1}a^{-1}\right)=e\Rightarrow ab=\left(b^{-1}a^{-1}\right)$ +\end_inset + + po definiciji inverza. +\end_layout + +\begin_layout Corollary* +Če so +\begin_inset Formula $a_{1},\dots,a_{k}$ +\end_inset + + obrnljivi elementi monoida, je +\begin_inset Formula $\left(a_{1}\cdots a_{k}\right)$ +\end_inset + + obrnljiv element monoida in velja +\begin_inset Formula $\left(a_{1}\cdots a_{k}\right)^{-1}=a_{k}^{-1}\cdots a_{1}^{-1}$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Dokaz z indukcijo: Baza: +\begin_inset Formula $k=2$ +\end_inset + + velja, Korak: predpostavimo +\begin_inset Formula $\left(a_{1}\cdots a_{k}\right)^{-1}=a_{k}^{-1}\cdots a_{1}^{-1}$ +\end_inset + +. + Množimo z desne z +\begin_inset Formula $a_{k+1}$ +\end_inset + + in smo dokazali. +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $\left(A,\cdot\right)$ +\end_inset + + monoid, +\begin_inset Formula $n\in\mathbb{N}_{0}$ +\end_inset + +. + +\begin_inset Formula $a^{0}\coloneqq e$ +\end_inset + +, +\begin_inset Formula $a^{n}=a\cdot a^{n-1}$ +\end_inset + + za +\begin_inset Formula $n\geq1$ +\end_inset + +. +\end_layout + +\begin_layout Corollary* +Velja torej +\begin_inset Formula $a^{n}a^{m}=a^{n+m}$ +\end_inset + +, +\begin_inset Formula $\left(a^{n}\right)^{m}=a^{nm}$ +\end_inset + +, +\begin_inset Formula $\left(a^{-1}\right)^{n}=a^{-1}\cdots n-\text{krat}\cdots a^{-1}=\left(a\cdots n-\text{krat}\cdots a\right)^{-1}=\left(a^{n}\right)^{-1}$ +\end_inset + +. + Torej za obrnljiv +\begin_inset Formula $a$ +\end_inset + + velja +\begin_inset Formula $\left(a^{n}\right)^{-1}=\left(a^{-1}\right)^{n}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +dodaj podgrupe etc +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Kaj je red elementa v grupi? Kaj je pravilo krajšanja v grupi? Dokažite + ga. + Kako se pravilo krajšanja odraža v Cayleyevi tabeli grupe? +\end_layout + +\begin_layout Definition* +Cayleyeva tabela za grupoid +\begin_inset Formula $\left(A,\cdot\right)$ +\end_inset + + je kvadratna tabela širine +\begin_inset Formula $\left|A\right|$ +\end_inset + +, kjer ima +\begin_inset Formula $i,j-$ +\end_inset + +ta celica vrednost +\begin_inset Formula $a_{i}\cdot a_{j}$ +\end_inset + +, kjer je +\begin_inset Formula $a_{k}$ +\end_inset + + +\begin_inset Formula $k-$ +\end_inset + +ti element množice +\begin_inset Formula $A$ +\end_inset + +. + Pač izberemo si neko linearno urejenost +\begin_inset Formula $A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $\left(G,\cdot\right)$ +\end_inset + + končna grupa in +\begin_inset Formula $a\in G$ +\end_inset + +. + Tedaj je red elementa +\begin_inset Formula $a$ +\end_inset + + najmanjši +\begin_inset Formula $n\in\mathbb{N}\ni:a^{n}=e$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Dirichletovo načelo: +\begin_inset Formula $\exists n,m\in\left[k+1\right],n\not=m:a^{n}=a^{m}$ +\end_inset + + +\end_layout + +\begin_layout Proof +BSŠ +\begin_inset Formula $n<m$ +\end_inset + +. + +\begin_inset Formula $a^{n}=a^{m}\Rightarrow a^{n}\left(a^{m}\right)^{-1}=a^{m}\left(a^{m}\right)^{-1}\Rightarrow a^{n-m}=e$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Theorem* +Pravilo krajšanja: Če je +\begin_inset Formula $G$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +Ko omenimo algebrajsko strukturo, občasno omenimo le množico, ko je iz konteksta + operacija implicitno znana. +\end_layout + +\end_inset + + grupa, +\begin_inset Formula $a,b,c\in G$ +\end_inset + +, velja +\begin_inset Formula $ab=ac\Rightarrow b=c$ +\end_inset + + in +\begin_inset Formula $ba=ca\Rightarrow b=c$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Množimo obe strani z leve ali desne z inverzom +\begin_inset Formula $a$ +\end_inset + +. +\end_layout + +\begin_layout Corollary* +V Cayleyevi tabeli grupe se v vsaki vrstici in v vsakem stolpcu pojavijo + vsi elementi grupe. +\end_layout + +\begin_layout Subsection +Kaj je permutacijska grupa? Kaj je simetrična grupa +\begin_inset Formula $S_{n}$ +\end_inset + + in kaj je alternirajoča grupa +\begin_inset Formula $A_{n}$ +\end_inset + +? Kaj pravi Cayleyev izrek (o univerzalnosti permutacijskih grup) in kakšna + je ideja njegovega dokaza? +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $A$ +\end_inset + + množica. + Permutacija +\begin_inset Formula $A$ +\end_inset + + je bijekcija +\begin_inset Formula $A\to A$ +\end_inset + +. + Permutacijska grupa je množica nekaj permutacij +\begin_inset Formula $A$ +\end_inset + +, ki ga komponiranje tvorijo grupo. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Simetrična grupa: +\begin_inset Formula $S_{n}\coloneqq\left\{ \pi;\pi:\left\{ 1..n\right\} \to\left\{ 1..n\right\} \text{ bijekcija}\right\} $ +\end_inset + +. + Alternirajoča grupa: +\begin_inset Formula $A_{n}\coloneqq\left\{ \pi\in S_{n};\pi\text{ soda}\right\} $ +\end_inset + +. + Permutacija je soda, če je v zapisu z disjunktnimi cikli vsota dolžin ciklov, + ki ji odštejemo število ciklov, soda. +\end_layout + +\begin_layout Theorem* +\begin_inset Formula $\left|A_{n}\right|=\frac{n!}{2}$ +\end_inset + +. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +DOKAŽI TODO XXX FIXME +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Definition* +Naj bosta +\begin_inset Formula $\left(G,\circ\right)$ +\end_inset + + in +\begin_inset Formula $\left(H,*\right)$ +\end_inset + + grupi. + +\begin_inset Formula $f:G\to H$ +\end_inset + + je hm +\begin_inset Formula $\varphi\Leftrightarrow\forall x,y\in G:f\left(x\circ y\right)=fx*fy$ +\end_inset + +. + Če je +\begin_inset Formula $f$ +\end_inset + + tudi bijekcija, je +\begin_inset Formula $f:G\to H$ +\end_inset + + izomorfizem. + Grupi sta izomorfni, če obstaja izomorfizem med njima. + Tedaj pišemo +\begin_inset Formula $G\approx H$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Theorem* +Cayley: Vsaka grupa je izomorfna neki permutacijski grupi. +\end_layout + +\begin_layout Proof +(skica dokaza) Naj bo +\begin_inset Formula $\left(G,\cdot\right)$ +\end_inset + + grupa. + Vsakemu elementu grupe priredimo preslikavo +\begin_inset Formula $g\in G\mapsto\alpha_{g}:G\to G$ +\end_inset + +, ki slika takole: +\begin_inset Formula $\forall x\in G:\alpha_{k}x=g\cdot x$ +\end_inset + +. + +\begin_inset Formula $\alpha_{g}$ +\end_inset + + je permutacija +\begin_inset Formula $G$ +\end_inset + +, saj je bijektivna, ker velja pravilo krajšanja: +\begin_inset Formula $x\not=y\Rightarrow\alpha_{g}x\overset{\text{pravilo krajšanja}}{=}gx\not=gy=\alpha_{g}y$ +\end_inset + +. + +\begin_inset Formula $\left\{ \alpha_{g};\forall g\in G\right\} $ +\end_inset + + tvori permutacijsko grupo, ki je izomorfna izvorni grupi: +\begin_inset Formula $\left(\left\{ \alpha_{g};\forall g\in G\right\} ,\circ\right)\approx\left(G,\cdot\right)$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Kaj so odseki grupe +\begin_inset Formula $G$ +\end_inset + + po podgrupi +\begin_inset Formula $H$ +\end_inset + +? Kdaj je +\begin_inset Formula $aH=H$ +\end_inset + + in kdaj je +\begin_inset Formula $aH=Ha$ +\end_inset + +? Kaj je vsebina Lagrangeovega izreka o moči podgrup in končnih grup? +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $\left(G,\cdot\right)$ +\end_inset + + grupa in +\begin_inset Formula $H$ +\end_inset + + podgrupa +\begin_inset Note Note +status open + +\begin_layout Plain Layout +prosim definiraj podgrupo +\end_layout + +\end_inset + + +\begin_inset Formula $G$ +\end_inset + +. + Če je +\begin_inset Formula $a\in G$ +\end_inset + +, definiramo +\begin_inset Formula $aH=\left\{ ah,h\in H\right\} $ +\end_inset + + (desni odsek podgrupe +\begin_inset Formula $H$ +\end_inset + +) in +\begin_inset Formula $Ha=\left\{ ha,h\in H\right\} $ +\end_inset + + (levi odsek podgrupe +\begin_inset Formula $H$ +\end_inset + +). +\end_layout + +\begin_layout Theorem* +Naj bo +\begin_inset Formula $G$ +\end_inset + + grupa in +\begin_inset Formula $a,b\in G$ +\end_inset + + in +\begin_inset Formula $H$ +\end_inset + + podgrupa +\begin_inset Formula $G$ +\end_inset + +. + Tedaj veljajo naslednje lastnosti: +\end_layout + +\begin_deeper +\begin_layout Enumerate +\begin_inset CommandInset label +LatexCommand label +name "enu:" + +\end_inset + + +\begin_inset Formula $a\in aH$ +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset CommandInset label +LatexCommand label +name "enu:-1" + +\end_inset + + +\begin_inset Formula $aH=H\Leftrightarrow a\in H$ +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $aH=bH\nabla aH\cap bH=\emptyset$ +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $aH=bH\Leftrightarrow a^{-1}b\in H$ +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\left|aH\right|=\left|bH\right|$ +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset CommandInset label +LatexCommand label +name "enu:-2" + +\end_inset + + +\begin_inset Formula $aH=Ha\Leftrightarrow H=aHa^{-1}$ +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $aH$ +\end_inset + + podgrupa +\begin_inset Formula $G\Leftrightarrow a\in H$ +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Proof +Dokažimo trditve: +\end_layout + +\begin_deeper +\begin_layout Enumerate +\begin_inset Formula $e\in H\Rightarrow a\cdot e=a\in aH$ +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + +\begin_inset Formula $a\overset{\ref{enu:}}{\in}aH=H\Rightarrow a\in H$ +\end_inset + +, +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + + Najprej ena vsebovanost: +\begin_inset Formula $\forall x\in aH\exists h\in H\ni:x=ah$ +\end_inset + +, ker +\begin_inset Formula $a\in H$ +\end_inset + +, je po zaprtosti podgrupe +\begin_inset Formula $H$ +\end_inset + + +\begin_inset Formula $ah=x\in H\Rightarrow aH\subseteq H$ +\end_inset + +. + Nato še druga vsebovanost: +\begin_inset Formula $\forall x\in H:a^{-1}x\in H\Rightarrow a\left(a^{-1}x\right)=x\in aH\Rightarrow H\subseteq aH$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Imejmo +\begin_inset Formula $aH$ +\end_inset + +, +\begin_inset Formula $bH$ +\end_inset + +. + Če je +\begin_inset Formula $aH=bH$ +\end_inset + +, smo dokazali, sicer +\begin_inset Formula $\exists x\in aH\cup bH$ +\end_inset + +. + Ker +\begin_inset Formula $x\in aH\Rightarrow\exists h'\in H\ni:x=ah'$ +\end_inset + +. + Ker +\begin_inset Formula $x\in bH\exists h''\in H\ni:x=bh''$ +\end_inset + +. + Toree velja +\begin_inset Formula $x=ah'=bh''$ +\end_inset + +. + Množimo s +\begin_inset Formula $h'^{-1}\Rightarrow a=bh''h'^{-1}\Rightarrow aH=bh''\left(h'^{-1}H\right)\overset{\ref{enu:-1}}{=}bh''H\overset{\ref{enu:-1}}{=}bH$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +... +\end_layout + +\begin_layout Enumerate +Očitno (pravilo krajšanja) +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + +\begin_inset Formula $aH=Ha\Rightarrow aHa^{-1}=Haa^{-1}=H$ +\end_inset + +. + +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + + +\begin_inset Formula $H=aHa^{-1}\overset{\cdot a^{-1}}{\Rightarrow}a^{-1}H=Ha$ +\end_inset + +. + Ker je +\begin_inset Formula $a\in aH$ +\end_inset + +, je v +\begin_inset Formula $a^{-1}\in aH$ +\end_inset + + (grupa), ker je +\begin_inset Formula $a^{-1}\in aH\cap a^{-1}H$ +\end_inset + +, je +\begin_inset Formula $aH=a^{-1}H$ +\end_inset + +, torej +\begin_inset Formula $a^{-1}H=Ha\Rightarrow aH=Ha$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Theorem* +Lagrange: Če je +\begin_inset Formula $G$ +\end_inset + + končna grupa in +\begin_inset Formula $H$ +\end_inset + + podgrupa +\begin_inset Formula $G$ +\end_inset + +, potem +\begin_inset Formula $\left|H\right|$ +\end_inset + + deli +\begin_inset Formula $\left|G\right|$ +\end_inset + +. + Nadalje je število levih/desnih odsekov po +\begin_inset Formula $H$ +\end_inset + + enako +\begin_inset Formula $\frac{\left|G\right|}{\left|H\right|}$ +\end_inset + +. +\end_layout + +\begin_layout Proof +Naj bodo +\begin_inset Formula $a_{1}H,a_{2}H,\dots,a_{n}H$ +\end_inset + + različni odseki. + Tedaj +\begin_inset Formula $\left|G\right|\overset{\ref{enu:}}{=}\left|a_{1}H\cup\cdots\cup a_{n}H\right|\overset{\text{disjunktni odseki}}{=}\sum_{i=1}^{k}\left|a_{i}H\right|=k\left|H\right|\Rightarrow k=\frac{\left|G\right|}{\left|H\right|}$ +\end_inset + +. +\end_layout + +\begin_layout Corollary* +Če je +\begin_inset Formula $G$ +\end_inset + + končna grupa in +\begin_inset Formula $a\in G$ +\end_inset + +, potem +\begin_inset Formula $\red a$ +\end_inset + + deli +\begin_inset Formula $\left|G\right|$ +\end_inset + +. +\end_layout + +\begin_layout Proof +\begin_inset Formula $\left\langle a\right\rangle =\left\{ a^{k};k\in\mathbb{Z}\right\} =\left\{ e,a,\dots,a^{n-1}\right\} \overset{\text{Lagrange}}{\Rightarrow}\left\langle a\right\rangle $ +\end_inset + + je gotovo podgrupa +\begin_inset Formula $G\Rightarrow n$ +\end_inset + + deli +\begin_inset Formula $\left|G\right|$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Kaj je podgrupa edinka? Navedite nekaj primerov podgrup edink. + Kaj je faktorska grupa +\begin_inset Formula $G/H$ +\end_inset + + in kaj pomeni, da je operacija v +\begin_inset Formula $G/H$ +\end_inset + + dobro definirana? +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $H$ +\end_inset + + podgrupa grupe +\begin_inset Formula $G$ +\end_inset + +. + Tedaj rečemo, da je +\begin_inset Formula $H$ +\end_inset + + podgrupa edinka, če velja +\begin_inset Formula $\forall a\in G:aH=Ha\overset{\ref{enu:-2}}{\Leftrightarrow}\forall a\in G:aHa^{-1}=H$ +\end_inset + +. + Oznaka: +\begin_inset Formula $H\triangleleft G$ +\end_inset + +. + Če sta +\begin_inset Formula $G$ +\end_inset + + in +\begin_inset Formula $\left\{ e\right\} $ +\end_inset + +, kjer je +\begin_inset Formula $e$ +\end_inset + + enota v +\begin_inset Formula $G$ +\end_inset + +, edini edinki +\begin_inset Formula $G$ +\end_inset + +, je +\begin_inset Formula $G$ +\end_inset + + enostavna. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Center grupe +\begin_inset Formula $G\sim ZG\coloneqq\left\{ a\in G;\forall x\in G:ax=xa\right\} $ +\end_inset + + ZDB taki elementi +\begin_inset Formula $G$ +\end_inset + +, ki komutirajo z vsemi. +\end_layout + +\begin_layout Example* +V Abelovi grupi je vsaka podgrupa edinka. + +\begin_inset Formula $SL_{2}\mathbb{R}\triangleleft GL_{2}\mathbb{R}$ +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +NAPIŠI KAJ JE TO SPECIAL LINEAR ITD +\end_layout + +\end_inset + +. + +\begin_inset Formula $ZG\triangleleft G$ +\end_inset + +. +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $H\triangleleft G$ +\end_inset + +. + +\begin_inset Formula $G/H\coloneqq\left\{ aH;a\in G\right\} $ +\end_inset + +. + V množico +\begin_inset Formula $G/H$ +\end_inset + + vpeljemo operacijo +\begin_inset Formula $\left(aH\right)\left(bH\right)=\left(ab\right)H$ +\end_inset + + (*). +\end_layout + +\begin_layout Theorem* +Faktorske grupe: Če je +\begin_inset Formula $H\triangleleft G$ +\end_inset + +, je +\begin_inset Formula $G/H$ +\end_inset + + grupa za (*). +\end_layout + +\begin_layout Proof +Dokazati notranjost, enoto, asociativnost in inverze je trivialno. + Treba je še dokazati dobro definiranost, t. + j. + +\begin_inset Formula $a,a'$ +\end_inset + + iz istega odseka in +\begin_inset Formula $b,b'$ +\end_inset + + iz istega odseka +\begin_inset Formula $\Rightarrow\left(aH\right)\left(bH\right)=\left(ab\right)H=\left(a'b'\right)H=\left(a'H\right)\left(b'H\right)$ +\end_inset + +. + Ker +\begin_inset Formula $a'\in aH\Rightarrow\exists h'\in H\ni:ah'=a'$ +\end_inset + +. + Ker +\begin_inset Formula $b'\in bH\Rightarrow\exists h''\in H\ni:bh''=b'$ +\end_inset + +. + Sedaj +\begin_inset Formula $\left(a'H\right)\left(b'H\right)\overset{\text{def.}}{=}\left(a'b'\right)H=\left(ah'bh''\right)H=ah'b\left(h''H\right)\overset{\ref{enu:-1}}{=}ah'bH=ah'\left(bH\right)\overset{H\text{ edinka}}{=}ah'\left(Hb\right)=a\left(h'H\right)b\overset{\ref{enu:-1}}{=}aHb=\left(ab\right)H$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Kaj je kolobar, kaj je cel kolobar? Kaj je pravilo krajšanja v kolobarjih + in kakšna je povezava tega pravila s celimi kolobarji? Kaj velja za končne + cele kolobarje? +\end_layout + +\begin_layout Definition* +Bigrupoid +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + je kolobar, če je +\begin_inset Formula $\left(R,+\right)$ +\end_inset + + abelova grupa, +\begin_inset Formula $\left(R,\cdot\right)$ +\end_inset + + polgrupa in velja +\begin_inset Formula $\forall a,b,c\in R:a\cdot\left(b+c\right)=a\cdot b+a\cdot c\wedge\left(a+b\right)\cdot c=a\cdot c+b\cdot c$ +\end_inset + + (leva in desna distributivnost). + Kolobar je komutativen, če je +\begin_inset Formula $\left(R,\cdot\right)$ +\end_inset + + komutativna polgrupa. + +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + je kolobar z enoto, če je +\begin_inset Formula $\left(R,\cdot\right)$ +\end_inset + + monoid. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Direktna vsota kolobarjev: Naj bosta +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $S$ +\end_inset + + kolobarja. + +\begin_inset Formula $R\oplus S=R\times S$ +\end_inset + + je direktna vsota . + Definirajmo operaciji +\begin_inset Formula $\left(r,s\right)+'\left(r',s'\right)\coloneqq\left(r+r',s+s'\right)$ +\end_inset + +, +\begin_inset Formula $\left(r,s\right)\cdot'\left(r',s'\right)\coloneqq\left(r\cdot r',s\cdot s'\right)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Center kolobarja +\begin_inset Formula $\sim ZR\coloneqq\left\{ a\in R;\forall x\in R:ax=xa\right\} $ +\end_inset + +, ZDB vsi taki elementi +\begin_inset Formula $R$ +\end_inset + +, ki pri množenju s poljubnim elementom kolobarja komutirajo. +\end_layout + +\begin_layout Claim* +Če sta +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $S$ +\end_inset + + kolobarja, je +\begin_inset Formula $\left(R\oplus S,+',\cdot'\right)$ +\end_inset + + kolobar. + Nadalje: Če sta +\begin_inset Formula $R$ +\end_inset + + in +\begin_inset Formula $S$ +\end_inset + + komutativna kolobara, je tudi +\begin_inset Formula $\left(R\oplus S,+',\cdot'\right)$ +\end_inset + + komutativen kolobar. + Če sta z enoto, je tudi +\begin_inset Formula $\left(R\oplus S,+',\cdot'\right)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +DEFINIRAJ PODKOLOBAR +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Definition* +Če v kolobarju +\begin_inset Formula $R$ +\end_inset + + velja +\begin_inset Formula $a\cdot b=0$ +\end_inset + + +\begin_inset Foot +status open + +\begin_layout Plain Layout +0 je aditivna enota +\end_layout + +\end_inset + + in sta +\begin_inset Formula $a\not=0$ +\end_inset + + in +\begin_inset Formula $b\not=0$ +\end_inset + +, sta +\begin_inset Formula $a$ +\end_inset + + in +\begin_inset Formula $b$ +\end_inset + + delitelja niča. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +V kolobarju +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + velja pravilo krajšanja, če velja +\begin_inset Formula $\forall a,b,c\in R,a\not=0:ab=ac\Rightarrow b=c$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Definition* +Komutativen kolobar z enoto +\begin_inset Formula $1\not=0$ +\end_inset + + (ZDB multiplikativna enota ni enaka aditivni) brez deliteljev niča je +\series bold +cel +\series default +. +\end_layout + +\begin_layout Example* +\begin_inset Formula $\left(\mathbb{Z},+,\cdot\right)$ +\end_inset + + je cel, toda +\begin_inset Formula $\left(\mathbb{Z}_{6},+_{6},\cdot_{6}\right)$ +\end_inset + + ni cel, ker premore delitelje niča. +\end_layout + +\begin_layout Theorem* +Naj bo +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + komutativen kolobar z enoto +\begin_inset Formula $1\not=0$ +\end_inset + +. + Velja +\begin_inset Formula $R$ +\end_inset + + cel +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + v +\begin_inset Formula $R$ +\end_inset + + velja pravilo krajšanja. +\end_layout + +\begin_layout Proof +Dokazujemo ekvivalenco: +\end_layout + +\begin_deeper +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + Predpostavimo, da je +\begin_inset Formula $R$ +\end_inset + + cel. + Predpostavimo +\begin_inset Formula $ab=ac$ +\end_inset + + za poljuben +\begin_inset Formula $a\not=0$ +\end_inset + + in poljubna +\begin_inset Formula $b,c$ +\end_inset + +. + Računamo: +\begin_inset Formula $ab=ac\Leftrightarrow ab-ac=0\Leftrightarrow a\left(b-c\right)=0$ +\end_inset + +. + Ker je +\begin_inset Formula $R$ +\end_inset + + cel in +\begin_inset Formula $a\not=0$ +\end_inset + +, mora biti +\begin_inset Formula $b-c=0$ +\end_inset + +, sicer bi bila +\begin_inset Formula $a$ +\end_inset + + in +\begin_inset Formula $b-c$ +\end_inset + + delitelja niča, kar bi bilo v +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + + s predpostavko o celosti +\begin_inset Formula $R$ +\end_inset + +. + +\begin_inset Formula $b-c=0\Rightarrow b=c$ +\end_inset + +, torej +\begin_inset Formula $\forall a\not=0,b,c\in R:ab=ac\Rightarrow b=c\sim$ +\end_inset + + velja pravilo krajšanja. +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + + Predpostavimo, da v komutativnem kolobarju z enoto +\begin_inset Formula $1\not=0$ +\end_inset + + +\begin_inset Formula $R$ +\end_inset + + velja pravilo krajšanja. + Predpostavimo +\begin_inset Formula $ab=0$ +\end_inset + +, +\begin_inset Formula $a\not=0$ +\end_inset + +. + Dokazati je treba +\begin_inset Formula $b=0$ +\end_inset + +, sicer bi imeli delitelje niča. + Velja +\begin_inset Formula $ab=0=a\cdot0$ +\end_inset + +. + Uporabimo pravilo krajšanja na +\begin_inset Formula $ab=a0$ +\end_inset + + in dobimo +\begin_inset Formula $b=0$ +\end_inset + +. + Kolobar je cel. +\end_layout + +\end_deeper +\begin_layout Definition* +Komutativen kolobar +\begin_inset Formula $R$ +\end_inset + + z enoto +\begin_inset Formula $1\not=0$ +\end_inset + + je +\series bold +obseg +\series default +, če je vsak neničeln element obrnljiv v +\begin_inset Formula $\left(R,\cdot\right)$ +\end_inset + + ZDB +\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$ +\end_inset + + je grupa. + Obseg +\begin_inset Formula $R$ +\end_inset + + je polje, če je +\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$ +\end_inset + + abelova grupa ZDB +\begin_inset Formula $ZR=R$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Če je +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + končen cel kolobar +\begin_inset Formula $\Rightarrow R$ +\end_inset + + polje. +\end_layout + +\begin_layout Proof +Dokažimo, da +\begin_inset Formula $\forall a\in R,a\not=0\exists a^{-1}\ni:aa^{-1}=1$ +\end_inset + +. + Naj bo +\begin_inset Formula $a\in R$ +\end_inset + + poljuben, z izjemo +\begin_inset Formula $a\not=0$ +\end_inset + +. + Oglejmo si +\begin_inset Formula $\left\{ a^{k};\forall k\geq0\right\} $ +\end_inset + + ZDB množico vseh potenc +\begin_inset Formula $a$ +\end_inset + +. + Ker +\begin_inset Formula $\left|R\right|<\infty\Rightarrow\exists i,j,$ +\end_inset + +BSŠ +\begin_inset Formula $i>j\ni a^{i}=a^{j}$ +\end_inset + + ZDB ker je +\begin_inset Formula $R$ +\end_inset + + končen, se nek element +\begin_inset Quotes gld +\end_inset + +ponovi +\begin_inset Quotes grd +\end_inset + +. + +\begin_inset Formula $a^{j}\cdot a^{i-j}=a^{i}=a^{j}=a^{j}\cdot1$ +\end_inset + + Ker je kolobar cel, +\begin_inset Formula $a^{j}\not=0$ +\end_inset + + in ker velja pravilo krajšanja, +\begin_inset Formula $a^{i-j}=1$ +\end_inset + +, pri čemer vemo, da je +\begin_inset Formula $i-j>0$ +\end_inset + +. + Ločimo dva primera: +\end_layout + +\begin_deeper +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $i-j=1$ +\end_inset + + +\begin_inset Formula $a=1\Rightarrow a$ +\end_inset + + je kot multiplikativna enota multiplikativni inverz sam sebi +\end_layout + +\begin_layout Labeling +\labelwidthstring 00.00.0000 +\begin_inset Formula $i-j>1$ +\end_inset + + +\begin_inset Formula $a=a\cdot a^{i-j-1}=1$ +\end_inset + +, torej +\begin_inset Formula $a^{i-j-1}$ +\end_inset + + je inverz +\begin_inset Formula $a$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Subsection +Kaj je karakteristika kolobarja? Kako lahko določimo karakteristiko kolobarja + z enoto? Kaj lahko povemo o karakteristiki celega kolobarja? +\end_layout + +\begin_layout Definition* +Karakteristika kolobarja +\begin_inset Formula $R\sim\karakteristika R$ +\end_inset + + je najmanjši +\begin_inset Formula $n\in\mathbb{N}\ni:\forall a\in R:a+\cdots_{n-\text{krat}}\cdots+a=0$ +\end_inset + +. + Če tak +\begin_inset Formula $n$ +\end_inset + + ne obstaja, pravimo +\begin_inset Formula $\karakteristika R=0$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Naj bo +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + kolobar z enoto. + +\begin_inset Formula $\karakteristika R=\red1$ +\end_inset + + v grupi +\begin_inset Formula $\left(R,+\right)$ +\end_inset + + ZDB red enote v aditivni grupi. +\end_layout + +\begin_layout Proof +Po definiciji reda je +\begin_inset Formula $1+\cdots_{\red1-\text{krat}}\cdots+1=0$ +\end_inset + + in za nek +\begin_inset Formula $m<\red1$ +\end_inset + + +\begin_inset Formula $1+\cdots_{m-\text{krat}}\cdots+1\not=0$ +\end_inset + +, torej +\begin_inset Formula $\karakteristika R\geq\red1$ +\end_inset + +. + Sedaj vzemimo poljuben +\begin_inset Formula $a\in R$ +\end_inset + +. + Velja +\begin_inset Formula $a+\cdots_{\red1-\text{krat}}\cdots+a=1\cdot a+\cdots_{\red1-\text{krat}}\cdots+1\cdot a=a\cdot\left(1+\cdots_{\red1-\text{krat}}\cdots+1\right)=a\cdot0=0$ +\end_inset + +, torej +\begin_inset Formula $\karakteristika R\leq\red1$ +\end_inset + +, torej +\begin_inset Formula $\karakteristika R=\red1$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Naj bo +\begin_inset Formula $\left(R,+,\cdot\right)$ +\end_inset + + cel kolobar. + Tedaj velja bodisi +\begin_inset Formula $\karakteristika R=0$ +\end_inset + + bodisi +\begin_inset Formula $\karakteristika R=p$ +\end_inset + +, kjer je +\begin_inset Formula $p$ +\end_inset + + praštevilo. +\end_layout + +\begin_layout Proof +Če je +\begin_inset Formula $\karakteristika R=0$ +\end_inset + +, smo dokazali, sicer je +\begin_inset Formula $\karakteristika R=r>0$ +\end_inset + +. + PDDRAA +\begin_inset Formula $n=pq$ +\end_inset + + za +\begin_inset Formula $p,q>1$ +\end_inset + + in +\begin_inset Formula $p,q\in\mathbb{N}$ +\end_inset + +. + Po prejšnjem izreku +\begin_inset Formula $\karakteristika R=\red1=n$ +\end_inset + +. + Tedaj velja +\begin_inset Formula $0=1+\cdots_{n-\text{krat}}\cdots+1=1+\cdots_{pq-\text{krat}}\cdots+1=\left(1+\cdots_{p-\text{krat}}\cdots+1\right)\cdot\left(1+\cdots_{q-\text{krat}}\cdots+1\right)=a\cdot b$ +\end_inset + +. + Ker smo v celem kolobarju, je bodisi +\begin_inset Formula $a=0$ +\end_inset + +, bodisi +\begin_inset Formula $b=0$ +\end_inset + +, toda to je +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +, saj bi bil potem +\begin_inset Formula $\red1=q$ +\end_inset + + ali +\begin_inset Formula $\red1=p$ +\end_inset + +, oboje pa je +\begin_inset Formula $<pq$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Kaj je ideal kolobarja? Kako lahko preverimo, ali je +\begin_inset Formula $I\subseteq R$ +\end_inset + + ideal kolobarja +\begin_inset Formula $R$ +\end_inset + +? Kaj je faktorski kolobar +\begin_inset Formula $R/I$ +\end_inset + + in kako računamo v njem? +\end_layout + +\begin_layout Definition* +Če je +\begin_inset Formula $R$ +\end_inset + + kolobar, je njegov podkolobar +\begin_inset Formula $S$ +\end_inset + + ideal, če velja +\begin_inset Formula $\forall v\in R,s\in S:rs,sr\in S$ +\end_inset + + ZDB to je podkolobar, zaprt za zunanje množenje. +\end_layout + +\begin_layout Example* +\begin_inset Formula $n\cdot\mathbb{Z}$ +\end_inset + + (večkratniki +\begin_inset Formula $n$ +\end_inset + +) za fiksen +\begin_inset Formula $n$ +\end_inset + + so ideal v +\begin_inset Formula $\mathbb{Z}$ +\end_inset + +. + +\begin_inset Formula $\mathbb{Z}$ +\end_inset + + je podkolobar v +\begin_inset Formula $\mathbb{Q}$ +\end_inset + +, toda ni njegov ideal. +\end_layout + +\begin_layout Claim* +\begin_inset Formula $I\subseteq R$ +\end_inset + + je ideal +\begin_inset Formula $\Leftrightarrow0\in I$ +\end_inset + + (vsebuje aditivno enoto) +\begin_inset Formula $\wedge\forall i,j\in I:i-j\in I$ +\end_inset + + (zaprt za odštevanje) +\begin_inset Formula $\wedge\forall i\in I,r\in R:ir\in I,ri\in I$ +\end_inset + + (zaprt za zunanje množenje) +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO XXX FIXME vsota in produkt idealov je spet ideal +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Definition* +Naj bo +\begin_inset Formula $R$ +\end_inset + + kolobar in +\begin_inset Formula $I$ +\end_inset + + ideal v +\begin_inset Formula $R$ +\end_inset + +. + Faktorski kolobar: +\begin_inset Formula $R/I=\left\{ a+I,\forall a\in R\right\} =\left\{ \left\{ a+i;\forall i\in I\right\} ;\forall a\in R\right\} $ +\end_inset + + vsebuje aditivne odseke. + V +\begin_inset Formula $R/I$ +\end_inset + + vpeljemo operaciji: +\begin_inset Formula $\left(a+I\right)+'\left(b+I\right)=a+b+I$ +\end_inset + + in +\begin_inset Formula $\left(a+I\right)\cdot'\left(b+I\right)=a\cdot b+I$ +\end_inset + +. +\end_layout + +\begin_layout Theorem* +Če je +\begin_inset Formula $I$ +\end_inset + + ideal v +\begin_inset Formula $R$ +\end_inset + +, je +\begin_inset Formula $\left(R/I,+',\cdot'\right)$ +\end_inset + + kolobar. +\end_layout + +\end_body +\end_document diff --git a/šola/krožek/05-17.odp b/šola/krožek/05-17.odp Binary files differnew file mode 100644 index 0000000..244b2b0 --- /dev/null +++ b/šola/krožek/05-17.odp diff --git a/šola/krožek/funkcije.odp b/šola/krožek/funkcije.odp Binary files differnew file mode 100644 index 0000000..fe98438 --- /dev/null +++ b/šola/krožek/funkcije.odp diff --git a/šola/la/dn8/dokument.lyx b/šola/la/dn8/dokument.lyx new file mode 100644 index 0000000..c603fcc --- /dev/null +++ b/šola/la/dn8/dokument.lyx @@ -0,0 +1,1349 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\begin_preamble +\usepackage{siunitx} +\usepackage{pgfplots} +\usepackage{listings} +\usepackage{multicol} +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}} +\usepackage{amsmath} +\usepackage{tikz} +\newcommand{\udensdash}[1]{% + \tikz[baseline=(todotted.base)]{ + \node[inner sep=1pt,outer sep=0pt] (todotted) {#1}; + \draw[densely dashed] (todotted.south west) -- (todotted.south east); + }% +}% +\DeclareMathOperator{\Lin}{Lin} +\DeclareMathOperator{\rang}{rang} +\DeclareMathOperator{\sled}{sled} +\end_preamble +\use_default_options true +\begin_modules +enumitem +theorems-ams +\end_modules +\maintain_unincluded_children false +\language slovene +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification false +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 2cm +\topmargin 2cm +\rightmargin 2cm +\bottommargin 2cm +\headheight 2cm +\headsep 2cm +\footskip 1cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style german +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +Rešitev osme domače naloge Linearne Algebre +\end_layout + +\begin_layout Author + +\noun on +Anton Luka Šijanec +\end_layout + +\begin_layout Date +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +today +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Abstract +Za boljšo preglednost sem svoje rešitve domače naloge prepisal na računalnik. + Dokumentu sledi še rokopis. + Naloge je izdelala asistentka Ajda Lemut. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +newcommand +\backslash +euler{e} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Dokaži, da je +\begin_inset Formula $\left[\left(x,y,z\right),\left(u,v,w\right)\right]=2xu-yu-xv+2yv-zv-yw+zw$ +\end_inset + + skalarni produkt in ugotovi, ali je +\begin_inset Formula +\[ +A=\left[\begin{array}{ccc} +0 & 2 & -2\\ +0 & 1 & 0\\ +-1 & 2 & -1 +\end{array}\right] +\] + +\end_inset + + normalna preslikava glede na +\begin_inset Formula $\left[\cdot,\cdot\right]$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Paragraph +Rešitev +\end_layout + +\begin_layout Standard +Predpostavljam polje +\begin_inset Formula $\mathbb{R}$ +\end_inset + + in vektorski prostor +\begin_inset Formula $V=\mathbb{R}^{3}$ +\end_inset + +, saj v kompleksnem to ni skalarni produkt (protiprimer pozitivne definitnosti + je +\begin_inset Formula $\left[\left(1,1,1+i\right),\left(1,1,1+i\right)\right]=2$ +\end_inset + +). + +\begin_inset Formula $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$ +\end_inset + + je skalarni produkt, če zadošča naslednjim lastnostim. + Dokažimo jih za +\begin_inset Formula $\left[\cdot,\cdot\right]$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\forall v\in V:v\not=0\Rightarrow\langle v,v\rangle>0$ +\end_inset + + +\begin_inset Formula +\[ +\left[\left(x,y,z\right),\left(x,y,z\right)\right]=2x^{2}-2xy+2y^{2}-2yz+z^{2}=2\left(x^{2}-xy+y^{2}\right)-2yz+z^{2}= +\] + +\end_inset + + +\begin_inset Formula +\[ +=2\left(\left(x-\frac{y}{2}\right)^{2}+\frac{3y^{2}}{4}\right)-2yz+z^{2}=2\left(x-\frac{y}{2}\right)^{2}-\frac{3y^{2}}{4}-2yz+z^{2}= +\] + +\end_inset + + +\begin_inset Formula +\[ +=2\left(x-\frac{y}{2}\right)^{2}+\left(\frac{\sqrt{3}y}{\sqrt{2}}-\frac{z\sqrt{2}}{\sqrt{3}}\right)^{2}+\frac{z^{2}}{3}\geq0 +\] + +\end_inset + +Sedaj poiščimo ničle. + Fiksirajmo poljubna +\begin_inset Formula $y$ +\end_inset + +, +\begin_inset Formula $z$ +\end_inset + + in uporabimo obrazec za ničle kvadratne enačbe: +\begin_inset Formula +\[ +x_{1,2}=\frac{2y\pm\sqrt{4y^{2}-8\left(2y^{2}-2yz+z^{2}\right)}}{4} +\] + +\end_inset + +Iščemo pozitivne diskriminante. +\begin_inset Formula +\[ +4y^{2}-8\left(2y^{2}-2yz+z^{2}\right)=-12y^{2}+16yz-8z^{2}=4 +\] + +\end_inset + +Fiksirajmo poljuben +\begin_inset Formula $z$ +\end_inset + +. + Vodilni koeficient kvadratne enačbe je negativen. + Uporabimo obrazec: +\begin_inset Formula +\[ +y_{1,2}=\frac{-16z\pm\sqrt{256z^{2}-384z^{2}=-128z^{2}}}{-24} +\] + +\end_inset + +Diskriminanta je nenegativna +\begin_inset Formula $\Leftrightarrow z=0$ +\end_inset + +. + Torej +\begin_inset Formula $z=0$ +\end_inset + +, zato +\begin_inset Formula $y=0$ +\end_inset + + in tudi +\begin_inset Formula $x=0$ +\end_inset + + glede na obrazce. + Skalarni produkt je res pozitivno definiten. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\forall v,u\in V:\langle v,u\rangle=\langle u,v\rangle$ +\end_inset + + +\begin_inset Formula +\[ +\left[\left(u,v,w\right),\left(x,y,z\right)\right]=\left[\left(x,y,z\right),\left(u,v,w\right)\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +2ux-vx-uy-2vy-wy-vz+wz=2xu-yu-xv+2yv-zv-vz+wz +\] + +\end_inset + + +\begin_inset Formula +\[ +0=0 +\] + +\end_inset + +Skalarni produkt je res simetričen. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\forall\alpha_{1},\alpha_{2}\in\mathbb{C}\forall u_{1},u_{2},v\in V:\langle\alpha_{1}v_{1}+\alpha_{2}v_{2},v\rangle=\alpha_{1}\langle u_{1},v\rangle+\alpha_{2}\langle u_{2},v\rangle$ +\end_inset + + +\begin_inset Formula +\[ +\left[\alpha_{1}\left(x_{1},y_{1},z_{1}\right)+\alpha_{2}\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\left[\left(\alpha_{1}x_{1}+\alpha_{2}x_{2},\alpha_{1}y_{1}+\alpha_{2}y_{2},\alpha_{1}z_{1}+\alpha_{2}z_{2}\right),\left(u,v,w\right)\right]= +\] + +\end_inset + + +\begin_inset Formula +\[ +=2\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)u-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)u-\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)v+ +\] + +\end_inset + + +\begin_inset Formula +\[ ++2\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)v-\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)v-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)w+\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)w= +\] + +\end_inset + + +\begin_inset Formula +\[ +=2\alpha_{1}x_{1}u+2\alpha_{2}x_{2}u-\alpha_{1}y_{1}u-\alpha_{2}y_{2}u-\alpha_{1}x_{1}v-\alpha_{2}x_{2}v+ +\] + +\end_inset + + +\begin_inset Formula +\[ ++2\alpha_{1}y_{1}v+2\alpha_{2}y_{2}v-\alpha_{1}z_{1}v-\alpha_{2}z_{2}v-\alpha_{1}y_{1}w-\alpha_{2}y_{2}w+\alpha_{1}z_{1}w+\alpha_{2}z_{2}w= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\alpha_{1}\left(2x_{1}u-y_{1}u-x_{1}v+2y_{1}v-z_{1}v-y_{1}w+z_{1}w\right)+\alpha_{2}\left(2x_{2}u-y_{2}u-x_{2}v+2y_{2}v-z_{2}v-y_{2}w+z_{2}w\right)= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\alpha_{1}\left[\left(x_{1},y_{1},z_{1}\right),\left(u,v,w\right)\right]+\alpha_{2}\left[\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right] +\] + +\end_inset + +Skalarni produkt je res homogen in linearen v prvem faktorju. +\end_layout + +\begin_layout Standard +Po definiciji +\begin_inset Formula $A$ +\end_inset + + normalna +\begin_inset Formula $\Leftrightarrow A^{*}A=AA^{*}$ +\end_inset + +. + Izračunajmo torej matriko +\begin_inset Formula $A^{*}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Na predavanjih 2024-05-08 smo dokazali, da za vsak skalarni produkt +\begin_inset Formula $\left[u,v\right]$ +\end_inset + + obstaja taka ortogonalna ( +\begin_inset Formula $M^{*}=M^{-1}$ +\end_inset + +) pozitivno definitna matrika +\begin_inset Formula $M$ +\end_inset + +, da velja +\begin_inset Formula $\left[u,v\right]=\langle u,Mv\rangle$ +\end_inset + +, kjer je +\begin_inset Formula $\langle\cdot,\cdot\rangle$ +\end_inset + + standardni skalarni produkt. +\end_layout + +\begin_layout Itemize +Izpeljimo predpis za +\begin_inset Formula $A^{*}$ +\end_inset + + pri podani matriki +\begin_inset Formula $A$ +\end_inset + + in skalarnem produktu +\begin_inset Formula $\left[\cdot,\cdot\right]$ +\end_inset + + s pripadajočo matriko +\begin_inset Formula $M$ +\end_inset + +: +\begin_inset Formula +\[ +\left[A^{*}x,y\right]=\left[x,Ay\right]\text{, uporabimo prvo točko:} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle A^{*}x,My\right\rangle =\left\langle x,MAy\right\rangle \text{, pišimo \ensuremath{z=My}:} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle A^{*}x,z\right\rangle =\left\langle x,MAM^{-1}z\right\rangle \text{, upoštevajmo drugo točko:} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle A^{*}x,z\right\rangle =\left\langle M^{-1}A^{\square}Mx,z\right\rangle \text{, kjer je \ensuremath{A^{\square}} adjungacija \ensuremath{A} pri standardnem skalarnem produktu} +\] + +\end_inset + + +\begin_inset Formula +\[ +\Rightarrow A^{*}=M^{-1}A^{\square}M=M^{-1}\overline{A}^{T}M\overset{A\in M\left(\mathbb{R}\right)}{=}M^{-1}A^{T}M +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Potrebujemo še matriko skalarnega produkta. +\begin_inset Formula +\[ +\left\langle \left(x,y,z\right),M\left(u,v,w\right)\right\rangle =\left[\begin{array}{ccc} +x & y & z\end{array}\right]\left[\begin{array}{ccc} +m_{11} & m_{12} & m_{13}\\ +m_{21} & m_{22} & m_{23}\\ +m_{31} & m_{32} & m_{33} +\end{array}\right]\left[\begin{array}{c} +u\\ +v\\ +w +\end{array}\right]=\left[\begin{array}{ccc} +x & y & z\end{array}\right]\left[\begin{array}{c} +um_{11}+vm_{12}+wm_{13}\\ +um_{21}+vm_{22}+wm_{23}\\ +um_{31}+vm_{32}+wm_{33} +\end{array}\right]= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\begin{array}{ccccc} + & xum_{11} & xvm_{12} & xwm_{13} & +\\ ++ & yum_{21} & yvm_{22} & ywm_{23} & +\\ ++ & zum_{31} & zvm_{32} & zwm_{33} +\end{array}=\left[\left(x,y,z\right),\left(u,v,w\right)\right]=2xu-yu-xv+2yv-zv-yw+zw\text{, torej} +\] + +\end_inset + + +\begin_inset Formula +\[ +M=\left[\begin{array}{ccc} +2 & -1 & 0\\ +-1 & 2 & -1\\ +0 & -1 & 1 +\end{array}\right]\text{, njen inverz pa je }M^{-1}=\left[\begin{array}{ccc} +1 & 1 & 1\\ +1 & 2 & 2\\ +1 & 2 & 3 +\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Izračunamo +\begin_inset Formula $A^{*}$ +\end_inset + + po formuli +\begin_inset Formula $A^{*}=M^{-1}A^{T}M$ +\end_inset + + in preverimo +\begin_inset Formula $A^{*}A=AA^{*}$ +\end_inset + +: +\begin_inset Formula +\[ +A^{*}=\left[\begin{array}{ccc} +1 & 1 & 1\\ +1 & 2 & 2\\ +1 & 2 & 3 +\end{array}\right]\left[\begin{array}{ccc} +0 & 0 & -1\\ +2 & 1 & 2\\ +-2 & 0 & -1 +\end{array}\right]\left[\begin{array}{ccc} +2 & -1 & 0\\ +-1 & 2 & -1\\ +0 & -1 & 1 +\end{array}\right]=\left[\begin{array}{ccc} +-1 & 2 & -1\\ +-2 & 3 & -1\\ +-6 & 6 & -2 +\end{array}\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +A^{*}A=\left[\begin{array}{ccc} +1 & -2 & 3\\ +1 & -3 & 5\\ +2 & -10 & 14 +\end{array}\right]\not=\left[\begin{array}{ccc} +8 & -6 & 2\\ +-2 & 3 & -1\\ +3 & -2 & 1 +\end{array}\right]=AA^{*} +\] + +\end_inset + + +\begin_inset Formula $A$ +\end_inset + + ni normala matrika. +\end_layout + +\begin_layout Itemize +Da preverimo pravilnost matrike +\begin_inset Formula $A^{*}$ +\end_inset + +, lahko napravimo preizkus: +\begin_inset Float figure +placement H +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename sage.png + width 100col% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Preizkus s programom SageMath. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dokazati, da +\begin_inset Formula $A$ +\end_inset + + ni normalna, je moč še lažje. + Dokažemo lahko namreč, da eden izmed potrebnih pogojev za normalnost matrike + ni izpolnjen. + Na primer: +\begin_inset Formula $AA^{*}=A^{*}A\rightarrow A=PDP^{-1}$ +\end_inset + +, kjer je +\begin_inset Formula $P$ +\end_inset + + ortogonalna in +\begin_inset Formula $D$ +\end_inset + + diagonalna +\begin_inset Formula $\Rightarrow$ +\end_inset + + lastni vektorji +\begin_inset Formula $A$ +\end_inset + + tvorijo ortogonalno množico. +\end_layout + +\begin_layout Standard +Lastne vrednosti +\begin_inset Formula $A$ +\end_inset + + so (s kalkulatorjem) +\begin_inset Formula $\left\{ -2,1\right\} $ +\end_inset + +, kjer ima 1 algebrajsko večkratnost 2. + Lastni vektorji: +\begin_inset Formula +\[ +A-\left(-2\right)I=\left[\begin{array}{ccc} +2 & 2 & -2\\ +0 & 3 & 0\\ +-1 & 2 & 1 +\end{array}\right]\sim\left[\begin{array}{ccc} +2 & 2 & -2\\ +0 & 3 & 0\\ +0 & 3 & 0 +\end{array}\right]\sim\left[\begin{array}{ccc} +2 & 2 & -2\\ +0 & 3 & 0\\ +0 & 0 & 0 +\end{array}\right]\sim\left[\begin{array}{ccc} +2 & 0 & -2\\ +0 & 3 & 0\\ +0 & 0 & 0 +\end{array}\right]\Rightarrow x=z,y=0\Rightarrow v_{1}=\left(1,0,1\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +A-1I=\left[\begin{array}{ccc} +-1 & 2 & -2\\ +0 & 0 & 0\\ +-1 & 2 & -2 +\end{array}\right]\sim\left[\begin{array}{ccc} +-1 & 2 & -2\\ +0 & 0 & 0\\ +0 & 0 & 0 +\end{array}\right]\Rightarrow x=2y-2z\Rightarrow v_{2}=\left(2,1,0\right),\quad v_{3}=\left(-2,0,1\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\left[v_{1},v_{2}\right]=\left[\left(1,0,1\right),\left(2,1,0\right)\right]=4-0-1+0-1-0+0=2\not=0\Rightarrow v_{1}\not\perp v_{2}\Rightarrow A\text{ ni normalna} +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +Pokaži +\begin_inset Formula $A:V\to V$ +\end_inset + + je normalna +\begin_inset Formula $\Leftrightarrow AA^{*}-A^{*}A$ +\end_inset + + je pozitivno semidefinitna. +\end_layout + +\begin_deeper +\begin_layout Paragraph +Rešitev +\end_layout + +\begin_layout Itemize +Definiciji: +\end_layout + +\begin_deeper +\begin_layout Itemize +\begin_inset Formula $A:V\to V$ +\end_inset + + je normalna +\begin_inset Formula $\Leftrightarrow A^{*}A=AA^{*}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $A:V\to V$ +\end_inset + + je pozitivno semidefinitna +\begin_inset Formula $\Leftrightarrow A=A^{*}\wedge\forall v\in V:\left\langle Av,v\right\rangle \geq0$ +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Itemize +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + Po predpostavki velja +\begin_inset Formula $AA^{*}=A^{*}A\Rightarrow AA^{*}-A^{*}A=0$ +\end_inset + +. +\begin_inset Formula +\[ +AA^{*}-A^{*}A\overset{?}{=}\left(AA^{*}-A^{*}A\right)^{*}\Leftrightarrow0=0^{*} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle \left(AA^{*}-A^{*}A\right)v,v\right\rangle =\left\langle 0v,v\right\rangle \overset{\text{homogenost}}{=}0\left\langle v,v\right\rangle =0\geq0 +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + +Po predpostavki velja +\begin_inset Formula $\left(AA^{*}-A^{*}A\right)^{*}=AA^{*}-A^{*}A$ +\end_inset + + in +\begin_inset Formula $\forall v\in V:\left\langle \left(AA^{*}-A^{*}A\right)v,v\right\rangle \geq0$ +\end_inset + +. +\begin_inset Formula +\[ +\sled\left(AA^{*}-A^{*}A\right)=\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)\overset{\text{lastnost sledi}}{=}\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)=0 +\] + +\end_inset + +Sled +\begin_inset Formula $M$ +\end_inset + + je vsota lastnih vrednosti +\begin_inset Formula $M$ +\end_inset + +, torej je vsota lastnih vrednosti +\begin_inset Formula $\left(AA^{*}-A^{*}A\right)=0$ +\end_inset + +. + +\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow$ +\end_inset + + vse lastne vrednosti so nenegativne. + Iz teh dveh trditev sledi, da je vsaka lastna vrednost +\begin_inset Formula $AA^{*}-A^{*}A=0$ +\end_inset + +. + +\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow AA^{*}-A^{*}A$ +\end_inset + + normalna. + Normalne matrike je moč diagonalizirati v ortonormirani bazi: +\begin_inset Formula +\[ +AA^{*}-A^{*}A=PDP^{-1}\overset{\text{diagonalci so lastne vrednosti}}{=}P0P^{-1}=0 +\] + +\end_inset + + +\begin_inset Formula +\[ +AA^{*}=A^{*}A\Rightarrow A\text{ je normalna} +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +Naj bo +\begin_inset Formula $w_{1}=\left(1,1,1,1\right)$ +\end_inset + +, +\begin_inset Formula $w_{2}=\left(3,3,-1,-1\right)$ +\end_inset + + in +\begin_inset Formula $y=\left(6,0,2,0\right)$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Enumerate +Poišči ortonormirano bazo za +\begin_inset Formula $W=\Lin\left\{ w_{1},w_{2}\right\} $ +\end_inset + + glede na standardni skalarni produkt. +\end_layout + +\begin_layout Enumerate +Izrazi +\begin_inset Formula $y$ +\end_inset + + kot vsoto vektorja iz +\begin_inset Formula $W$ +\end_inset + + in vektorja iz +\begin_inset Formula $W^{\perp}$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph +Rešitev +\end_layout + +\begin_layout Enumerate +Uporabimo Gram-Schmidtov postopek in sproti normiramo bazne vektorje: +\begin_inset Formula +\[ +v_{1}=\left(1,1,1,1\right)/2=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\tilde{v_{2}}=\left(3,3,-1,-1\right)-\left\langle \left(3,3,-1,-1\right),v_{1}\right\rangle v_{1}=\left(3,3,-1,-1\right)-\left\langle \left(3,3,-1,-1\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\left(3,3,-1,-1\right)-2\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)=\left(2,2,-2,-2\right),\quad\quad\quad v_{2}=\tilde{v_{2}}/\left|\left|\tilde{v_{2}}\right|\right|=\text{\ensuremath{\tilde{v_{2}}/4}}=\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right) +\] + +\end_inset + +Baza za +\begin_inset Formula $W$ +\end_inset + + je +\begin_inset Formula $B=\left\{ v_{1},v_{2}\right\} =\left\{ \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\} $ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Dopolnimo +\begin_inset Formula $B$ +\end_inset + + do baze +\begin_inset Formula $\mathbb{R}^{4}$ +\end_inset + +. + Dopolnitev +\begin_inset Formula $\left\{ u_{1},u_{2}\right\} $ +\end_inset + + bo ortonormirana baza za +\begin_inset Formula $W^{\perp}$ +\end_inset + +, nato uporabimo Fourierov razvoj po dopolnjeni bazi. + Bazo podprostora dopolnimo tako, da rešimo sistem enačb. +\begin_inset Formula +\[ +\left\langle \left(x_{1},y_{1},z_{1},w_{1}\right),\left(3,3,-1,-1\right)\right\rangle =0\quad\quad\quad\left\langle \left(x_{2},y_{2},z_{2},w_{2}\right),\left(1,1,1,1\right)\right\rangle =0 +\] + +\end_inset + + +\begin_inset Formula +\[ +\left[\begin{array}{cccc} +1 & 1 & 1 & 1\\ +3 & 3 & -1 & -1 +\end{array}\right]\sim\left[\begin{array}{cccc} +1 & 1 & 0 & 0\\ +0 & 0 & 1 & 1 +\end{array}\right]\Rightarrow x=-y,\quad z=-w\Rightarrow\tilde{u_{1}}=\left(1,-1,0,0\right),\quad\tilde{u_{2}}=\left(0,0,1,-1\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +u_{1}=\tilde{u_{1}}/\left|\left|\tilde{u_{1}}\right|\right|=\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\quad\quad\quad u_{2}=\tilde{u_{2}}/\left|\left|\tilde{u}_{2}\right|\right|=\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}=\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle v_{1}+\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\rangle v_{2}+ +\] + +\end_inset + + +\begin_inset Formula +\[ ++\left\langle \left(6,0,2,0\right),\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\right\rangle u_{1}+\left\langle \left(6,0,2,0\right),\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},\right)\right\rangle u_{2}=4v_{1}+2v_{2}+\frac{6}{\sqrt{2}}u_{1}+\frac{2}{\sqrt{2}}u_{2}= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\left(2,2,2,2\right)+\left(1,1,-1,-1\right)+\left(3,-3,0,0\right)+\left(0,0,1,-1\right)=\left(3,3,1,1\right)\in W+\left(3,-3,1,-1\right)\in W^{\perp} +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +Poišči singularni razcep matrike +\begin_inset Formula +\[ +A=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & -2 & 0\\ +0 & 0 & 0\\ +0 & 0 & 0 +\end{array}\right]\text{.} +\] + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Paragraph +Rešitev +\end_layout + +\begin_layout Itemize +Iščemo +\begin_inset Formula $U$ +\end_inset + +, +\begin_inset Formula $\Sigma$ +\end_inset + + in +\begin_inset Formula $V$ +\end_inset + +, da velja +\begin_inset Formula $A=U\Sigma V^{*}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Diagonalci +\begin_inset Formula $\Sigma$ +\end_inset + + so singularne vrednosti +\begin_inset Formula $A$ +\end_inset + +. + Singularne vrednosti +\begin_inset Formula $A$ +\end_inset + + so koreni lastnih vrednosti +\begin_inset Formula $A^{*}A$ +\end_inset + +, torej +\begin_inset Formula $\sigma_{1}=2$ +\end_inset + +, +\begin_inset Formula $\sigma_{2}=1$ +\end_inset + +, +\begin_inset Formula $\sigma_{3}=0$ +\end_inset + +. +\begin_inset Formula +\[ +A^{*}A=\left[\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & -2 & 0 & 0\\ +0 & 0 & 0 & 0 +\end{array}\right]\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & -2 & 0\\ +0 & 0 & 0\\ +0 & 0 & 0 +\end{array}\right]=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & 4 & 0\\ +0 & 0 & 0 +\end{array}\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +\Sigma=\left[\begin{array}{ccc} +\sigma_{1} & 0 & 0\\ +0 & \sigma_{2} & 0\\ +0 & 0 & \sigma_{3}\\ +0 & 0 & 0 +\end{array}\right]=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & 2 & 0\\ +0 & 0 & 0\\ +0 & 0 & 0 +\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Stolpci +\begin_inset Formula $V$ +\end_inset + + so ortonormirana baza jedra +\begin_inset Formula $A^{*}A-\sigma^{2}I$ +\end_inset + + za vse singularne vrednosti +\begin_inset Formula $\sigma$ +\end_inset + +. +\begin_inset Formula +\[ +A^{*}A-4I=\left[\begin{array}{ccc} +-3 & 0 & 0\\ +0 & 0 & 0\\ +0 & 0 & -4 +\end{array}\right]\Rightarrow x=z=0\Rightarrow v_{1}=\left(0,1,0\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +A^{*}A-1I=\left[\begin{array}{ccc} +0 & 0 & 0\\ +0 & 3 & 0\\ +0 & 0 & -1 +\end{array}\right]\Rightarrow y=z=0\Rightarrow v_{2}=\left(1,0,0\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +A^{*}A-0I=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & 4 & 0\\ +0 & 0 & 0 +\end{array}\right]\Rightarrow x=y=0\Rightarrow v_{3}=\left(0,0,1\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +V=\left[\begin{array}{ccc} +v_{1} & v_{2} & v_{3}\end{array}\right]=\left[\begin{array}{ccc} +0 & 1 & 0\\ +1 & 0 & 0\\ +0 & 0 & 1 +\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Stolpci +\begin_inset Formula $U$ +\end_inset + + so ortonormirana baza in velja +\begin_inset Formula $\forall i\in\left\{ 1..\rang A\right\} :u_{i}=\sigma_{i}^{-1}Av_{i}$ +\end_inset + +. + Stolpične vektorje +\begin_inset Formula $v_{\rang A+1},\dots,v_{m}$ +\end_inset + + najdemo tako, da dopolnimo +\begin_inset Formula $v_{1},\dots,v_{\rang A}$ +\end_inset + + do ONB. +\begin_inset Formula +\[ +U=\left[\begin{array}{cccc} +0 & 1 & 0 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1\\ +0 & 0 & 1 & 0 +\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Dobljene matrike zmnožimo, s čimer potrdimo veljavnost singularnega razcepa: +\begin_inset Formula +\[ +U\Sigma V^{*}=\left[\begin{array}{cccc} +0 & 1 & 0 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1\\ +0 & 0 & 1 & 0 +\end{array}\right]\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & 2 & 0\\ +0 & 0 & 0\\ +0 & 0 & 0 +\end{array}\right]\left[\begin{array}{ccc} +0 & 1 & 0\\ +1 & 0 & 0\\ +0 & 0 & 1 +\end{array}\right]=\left[\begin{array}{ccc} +1 & 0 & 0\\ +0 & -2 & 0\\ +0 & 0 & 0\\ +0 & 0 & 0 +\end{array}\right]=A +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Standard +Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja. + Zavedam se namreč njihovega neličnega izgleda. +\end_layout + +\begin_layout Standard +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/1ladn8a.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/1ladn8aq.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/1ladn8b.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/1ladn8c.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/3ladn8.jpg + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/4ladn8.jpg + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/šola/la/dn8/sage.png b/šola/la/dn8/sage.png Binary files differnew file mode 100644 index 0000000..1cd41f5 --- /dev/null +++ b/šola/la/dn8/sage.png diff --git a/šola/la/kolokvij4.lyx b/šola/la/kolokvij4.lyx new file mode 100644 index 0000000..3e8a3e8 --- /dev/null +++ b/šola/la/kolokvij4.lyx @@ -0,0 +1,1068 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\begin_preamble +\usepackage{siunitx} +\usepackage{pgfplots} +\usepackage{listings} +\usepackage{multicol} +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}} +\end_preamble +\use_default_options true +\begin_modules +enumitem +\end_modules +\maintain_unincluded_children false +\language slovene +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification false +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 1cm +\topmargin 2cm +\rightmargin 1cm +\bottommargin 2cm +\headheight 1cm +\headsep 1cm +\footskip 1cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style german +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +newcommand +\backslash +euler{e} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{multicols}{2} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Paragraph +Drobnarije od prej +\end_layout + +\begin_layout Standard +\begin_inset Formula $\det A=\det A^{T}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Vsota je direktna +\begin_inset Formula $\Leftrightarrow V\cap U=\left\{ 0\right\} $ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Skalarni produkt +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left\langle v,v\right\rangle >0$ +\end_inset + +, +\begin_inset Formula $\left\langle v,u\right\rangle =\overline{\left\langle u,v\right\rangle }$ +\end_inset + +, +\begin_inset Formula $\left\langle \alpha_{2}u_{1}+\alpha_{2}u_{2},v\right\rangle =\alpha_{1}\left\langle u_{1},v\right\rangle +\alpha_{2}\left\langle u_{2},v\right\rangle $ +\end_inset + +, +\begin_inset Formula $\left\langle u,\alpha_{1}v_{1}+\alpha_{2}v_{2}\right\rangle =\overline{\alpha_{1}}\left\langle u,v_{1}\right\rangle +\overline{\alpha_{2}}\left\langle u,v_{2}\right\rangle $ +\end_inset + + +\end_layout + +\begin_layout Standard +Standardni: +\begin_inset Formula $\left\langle \left(\alpha_{1},\dots,\alpha_{n}\right),\left(\beta_{1},\dots,\beta_{n}\right)\right\rangle =\alpha_{1}\overline{\beta_{1}}+\cdots\alpha_{n}\overline{\beta_{n}}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Norma: +\begin_inset Formula $\left|\left|v\right|\right|^{2}=\left\langle v,v\right\rangle $ +\end_inset + +: +\begin_inset Formula $\left|\left|v\right|\right|>0\Leftrightarrow v\not=0$ +\end_inset + +, +\begin_inset Formula $\left|\left|\alpha v\right|\right|=\left|\alpha\right|\left|\left|v\right|\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +Trikotniška neenakost: +\begin_inset Formula $\left|\left|u+v\right|\right|\leq\left|\left|u\right|\right|+\left|\left|v\right|\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +Cauchy-Schwarz: +\begin_inset Formula $\left|\left\langle u,v\right\rangle \right|\leq\left|\left|v\right|\right|\left|\left|u\right|\right|$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $v\perp u\Leftrightarrow\left\langle u,v\right\rangle =0$ +\end_inset + +. + +\begin_inset Formula $M$ +\end_inset + + ortog. + +\begin_inset Formula $\Leftrightarrow\forall u,v\in M:v\perp u\wedge v\not=0$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $M$ +\end_inset + + normirana +\begin_inset Formula $\Leftrightarrow\forall u\in M:\left|\left|u\right|\right|=1$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $M$ +\end_inset + + ortog. + +\begin_inset Formula $\Rightarrow M$ +\end_inset + + lin. + neod., Ortog. + baza +\begin_inset Formula $\sim$ +\end_inset + + ortog. + ogrodje +\end_layout + +\begin_layout Standard +\begin_inset Formula $v\perp M\Leftrightarrow\forall u\in M:v\perp u$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Fourierov razvoj +\end_layout + +\begin_layout Standard +\begin_inset Formula $v_{i}$ +\end_inset + + ortog. + baza za +\begin_inset Formula $V$ +\end_inset + +, +\begin_inset Formula $v\in V$ +\end_inset + + poljuben. + +\begin_inset Formula $v=\sum_{i=1}^{n}\frac{\left\langle v,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Parsevalova identiteta: +\begin_inset Formula $\left|\left|v\right|\right|^{2}=\sum_{i=1}^{n}\frac{\left|\left\langle v,v_{i}\right\rangle \right|^{2}}{\left\langle v_{i},v_{i}\right\rangle }$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Projekcija na podprostor +\end_layout + +\begin_layout Standard +let +\begin_inset Formula $V$ +\end_inset + + podprostor +\begin_inset Formula $W$ +\end_inset + +. + +\begin_inset Formula $v'$ +\end_inset + + je ortog. + proj vektorja +\begin_inset Formula $v$ +\end_inset + + +\begin_inset Formula $\Leftrightarrow\forall w\in W:\left|\left|v-v'\right|\right|\leq\left|\left|v-w\right|\right|\sim\text{v'}$ +\end_inset + + je najbližje +\begin_inset Formula $V$ +\end_inset + + izmed elementov +\begin_inset Formula $W$ +\end_inset + +. + +\begin_inset Formula $\sun$ +\end_inset + + Pitagora: +\end_layout + +\begin_layout Standard +Zadošča preveriti ortogonalnost +\begin_inset Formula $v-v'$ +\end_inset + + na vse elemente +\begin_inset Formula $W$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Formula za ort. + proj.: +\begin_inset Formula $v'=\sum_{i=0}^{n}\frac{\left\langle v,w_{i}\right\rangle }{\left\langle w_{i},w_{i}\right\rangle }$ +\end_inset + +, kjer je +\begin_inset Formula $w_{i}$ +\end_inset + + OB +\begin_inset Formula $W$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph +Obstoj ortogonalne baze (Gram-Schmidt) +\end_layout + +\begin_layout Standard +let +\begin_inset Formula $\left\{ u_{1},\dots,u_{n}\right\} $ +\end_inset + + baza +\begin_inset Formula $V$ +\end_inset + +. + Zanj konstruiramo OB +\begin_inset Formula $\left\{ v_{1},\dots,v_{n}\right\} $ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $v_{1}=u_{1}$ +\end_inset + +, +\begin_inset Formula $v_{2}=u_{2}-\frac{\left\langle u_{2},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$ +\end_inset + +, +\begin_inset Formula $v_{3}=u_{3}-\frac{\left\langle u_{3},v_{2}\right\rangle }{\left\langle v_{2},v_{2}\right\rangle }v_{2}-\frac{\left\langle u_{3},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$ +\end_inset + +... + +\begin_inset Formula $v_{k}=u_{k}-\sum_{i=1}^{k-1}\frac{\text{\left\langle u_{k},v_{i}\right\rangle }}{\left\langle v_{i},v_{i}\right\rangle }v_{i}$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Ortogonalni komplement +\end_layout + +\begin_layout Standard +let +\begin_inset Formula $S\subseteq V$ +\end_inset + +. + +\begin_inset Formula $S^{\perp}=\left\{ v\in V;v\perp S\right\} $ +\end_inset + +. + Velja: +\begin_inset Formula $S^{\perp}$ +\end_inset + + podprostor +\begin_inset Formula $V$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $W$ +\end_inset + + podprostor +\begin_inset Formula $V$ +\end_inset + +. + Velja: +\begin_inset Formula $W\oplus W^{\perp}=V$ +\end_inset + + in +\begin_inset Formula $\left(W^{\perp}\right)^{\perp}=W$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Če je +\begin_inset Formula $\left\{ u_{1},\dots,u_{k}\right\} $ +\end_inset + + OB podprostora +\begin_inset Formula $V$ +\end_inset + +, je dopolnitev do baze vsega +\begin_inset Formula $V^{\perp}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Za vektorske podprostore +\begin_inset Formula $V_{i}$ +\end_inset + + VPSSP +\begin_inset Formula $W$ +\end_inset + + velja: +\end_layout + +\begin_layout Standard +\begin_inset Formula $S\subseteq W\Rightarrow\left(S^{\perp}\right)^{\perp}=\mathcal{L}in\left\{ S\right\} $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $V_{1}\subseteq V_{2}\Rightarrow V_{2}^{\perp}\subseteq V_{1}^{\perp}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(V_{1}+v_{2}\right)^{\perp}=V_{1}^{\perp}\cup V_{2}^{\perp}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(V_{1}\cap V_{2}\right)^{\perp}=V_{1}^{\perp}+V_{2}^{\perp}$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Linearni funkcional +\end_layout + +\begin_layout Standard +je linearna preslikava +\begin_inset Formula $V\to F$ +\end_inset + +, če je +\begin_inset Formula $V$ +\end_inset + + nad poljem +\begin_inset Formula $F$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Rieszov izrek o reprezentaciji linearnih funkcionalov: +\begin_inset Formula $\forall\text{l.f.}\varphi:V\to F\exists!w\in V\ni:\forall v\in V:\varphi v=\left\langle v,w\right\rangle $ +\end_inset + + +\end_layout + +\begin_layout Standard +Za +\begin_inset Formula $L:U\to V$ +\end_inset + + je +\begin_inset Formula $L^{*}:V\to U$ +\end_inset + + adjungirana linearna preslika +\begin_inset Formula $\Leftrightarrow\forall u\in U,v\in V:\left\langle Lu,v\right\rangle =\left\langle v,L^{*}u\right\rangle $ +\end_inset + + +\end_layout + +\begin_layout Standard +Za std. + skal. + prod. + velja: +\begin_inset Formula $A^{*}=\overline{A}^{T}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(AB\right)^{*}=B^{*}A^{*}$ +\end_inset + +, +\begin_inset Formula $\left(L^{*}\right)_{B\leftarrow C}=\left(L_{C\leftarrow B}\right)^{*}$ +\end_inset + +, +\begin_inset Formula $\left(\alpha A+\beta B\right)^{*}=\overline{\alpha}A^{*}+\overline{\beta}B^{*}$ +\end_inset + +, +\begin_inset Formula $\left(A^{*}\right)^{*}=A$ +\end_inset + +, +\begin_inset Formula $\text{Ker}L^{*}=\left(\text{Im}L\right)^{\perp}$ +\end_inset + +, +\begin_inset Formula $\left(\text{Ker}L^{*}\right)^{\perp}=\text{Im}L$ +\end_inset + +, +\begin_inset Formula $\text{Ker}\left(L^{*}L\right)=\text{Ker}L$ +\end_inset + +, +\begin_inset Formula $\text{Im}\left(L^{*}L\right)=\text{Im}L$ +\end_inset + + +\end_layout + +\begin_layout Standard +Lastne vrednosti +\begin_inset Formula $A^{*}$ +\end_inset + + so konjugirane lastne vrednosti +\begin_inset Formula $A$ +\end_inset + +. + Dokaz: +\begin_inset Formula $B=A-\lambda I$ +\end_inset + +. + +\begin_inset Formula $B^{*}=A^{*}-\overline{\lambda}I$ +\end_inset + +. + +\begin_inset Formula $\det B^{*}=\det\overline{B}^{T}=\det B=\overline{\det B}$ +\end_inset + +, torej +\begin_inset Formula $\det B=0\Leftrightarrow\det B^{*}=0$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\Delta_{A^{*}}$ +\end_inset + + ima konjugirane koeficiente +\begin_inset Formula $\Delta_{A}$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph +Normalne matrike +\begin_inset Formula $A^{*}A=AA^{*}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Velja: +\begin_inset Formula $A$ +\end_inset + + kvadratna, +\begin_inset Formula $Av=\lambda v\Leftrightarrow A^{*}v=\overline{\lambda}v$ +\end_inset + + (isti lastni vektorji) +\end_layout + +\begin_layout Standard +\begin_inset Formula $Au=\lambda u\wedge Av=\mu v\wedge\mu\not=\lambda\Rightarrow v\perp u$ +\end_inset + + +\end_layout + +\begin_layout Standard +Je podobna diagonalni: +\begin_inset Formula $\forall m:\text{Ker}\left(A-\lambda I\right)^{m}=\text{Ker}\left(A-\lambda I\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $A=PDP^{-1}\Leftrightarrow$ +\end_inset + + stolpci +\begin_inset Formula $P$ +\end_inset + + so ONB, diagonalci +\begin_inset Formula $D$ +\end_inset + + lavr, zdb +\begin_inset Formula $P$ +\end_inset + + je unitarna/ortogonalna. +\end_layout + +\begin_layout Paragraph +Unitarne +\begin_inset Formula $\mathbb{C}$ +\end_inset + +/ortogonalne +\begin_inset Formula $\mathbb{R}$ +\end_inset + + matrike +\begin_inset Formula $AA^{*}=A^{*}A=I$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $A$ +\end_inset + + kvadratna z ON stolpci. + +\begin_inset Formula $A$ +\end_inset + + ortog. + +\begin_inset Formula $\Rightarrow A$ +\end_inset + + normalna +\end_layout + +\begin_layout Standard +Lavr: let +\begin_inset Formula $Av=\lambda v\Rightarrow\left\langle Av,Av\right\rangle =\left\langle \lambda v,\lambda v\right\rangle =\left\langle v,v\right\rangle =\lambda\overline{\lambda}\left\langle v,v\right\rangle \Rightarrow\left|\lambda\right|=1\Rightarrow\lambda=e^{i\varphi}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $A=PDP^{-1},A^{*}=A^{-1}$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Simetrične +\begin_inset Formula $\mathbb{R}$ +\end_inset + +/hermitske +\begin_inset Formula $\mathbb{C}$ +\end_inset + + matrike +\begin_inset Formula $A=A^{*}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Sebiadjungirane linearne preslikave. +\end_layout + +\begin_layout Standard +Hermitska +\begin_inset Formula $\Rightarrow$ +\end_inset + + Normalna +\end_layout + +\begin_layout Standard +\begin_inset Formula $Av=\lambda v=A^{*}v=\overline{\lambda}v\Rightarrow\lambda\in\mathbb{R}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $A=A^{*}\Leftrightarrow\forall v:\left\langle Av,v\right\rangle \in\mathbb{R}$ +\end_inset + + +\end_layout + +\begin_layout Paragraph +Pozitivno (semi)definitne +\begin_inset Formula $A\geq0$ +\end_inset + + ( +\begin_inset Formula $>$ +\end_inset + + za PD) +\end_layout + +\begin_layout Standard +\begin_inset Formula $A$ +\end_inset + + P(S)D +\begin_inset Formula $\Rightarrow$ +\end_inset + + +\begin_inset Formula $A$ +\end_inset + + sim./ortog. + +\begin_inset Formula $\Rightarrow A$ +\end_inset + + normalna +\end_layout + +\begin_layout Standard +Def.: +\begin_inset Formula $A=A^{*}\wedge\forall v:\left\langle Av,v\right\rangle \geq0$ +\end_inset + + ( +\begin_inset Formula $>$ +\end_inset + + za PD) +\end_layout + +\begin_layout Standard +Za poljubno +\begin_inset Formula $B$ +\end_inset + + je +\begin_inset Formula $B^{*}B$ +\end_inset + + PSD. + Če ima +\begin_inset Formula $B$ +\end_inset + + LN stolpce, je +\begin_inset Formula $B^{*}B$ +\end_inset + + PD. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall\text{lavr}\lambda_{i}:A>0\Rightarrow\lambda_{i}>0$ +\end_inset + +, +\begin_inset Formula $A\geq0\Rightarrow\lambda_{i}\geq0$ +\end_inset + +. + Dokaz: let +\begin_inset Formula $A\geq0,v\not=0,Av=\lambda v\Rightarrow\left\langle Av,v\right\rangle =\left\langle \lambda v,v\right\rangle =\lambda\left\langle v,v\right\rangle \geq0\wedge\left\langle v,v\right\rangle >0\Rightarrow\lambda\geq0$ +\end_inset + + +\end_layout + +\begin_layout Standard +Lavr isto kot hermitska, lave isto kot normalna, diag. + isto kot normalna. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\text{A\ensuremath{\geq0}}\Rightarrow\exists B=B^{*},B\geq0\ni:B^{2}=A$ +\end_inset + +. + Dokaz: let +\begin_inset Formula $E\text{diag s koreni lavr}\geq0,A=PDP^{-1},P^{*}=P^{-1},D=\text{\text{diag z lavr}}\geq0,B=PEP^{-1}=PEP^{*}\Rightarrow B=B^{*}\Rightarrow B^{2}=PEP^{-1}PEP^{-1}=PE^{2}P^{-1}=PDP=A$ +\end_inset + + +\end_layout + +\begin_layout Standard +NTSE: +\begin_inset Formula $A\geq0\Leftrightarrow A=A^{*}\wedge\forall\lambda\text{lavr}A:\lambda\geq0\Leftrightarrow A=PDP^{-1}\wedge P\text{ unit.}\wedge\text{diag.}D\geq0\Leftrightarrow A=A^{*}\wedge\exists\sqrt{A}\ni:\sqrt{A}^{2}=A\Leftrightarrow A=B^{*}B$ +\end_inset + + (oz. + +\begin_inset Formula $>$ +\end_inset + + za PD) +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall\left[\cdot,\cdot\right]:V^{2}\to F\exists M>0\ni:\forall v,u\in V:\left[v,u\right]=\left\langle Au,v\right\rangle $ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\forall A>0:\left\langle A\cdot,\cdot\right\rangle $ +\end_inset + + je skalarni produkt. +\end_layout + +\begin_layout Paragraph +Singularni razcep (SVD) +\end_layout + +\begin_layout Standard +Singularne vrednosti +\begin_inset Formula $A$ +\end_inset + + so kvadratni koreni lastnih vrednosti +\begin_inset Formula $A^{*}A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Št. + ničelnih singvr +\begin_inset Formula $=\dim\text{Ker}\left(A^{*}A\right)=\dim\text{Ker}A$ +\end_inset + + +\end_layout + +\begin_layout Standard +Št. + nenič. + singvr +\begin_inset Formula $n\times n$ +\end_inset + + matrike +\begin_inset Formula $=n-\dim\text{Ker}A=\text{rang}A$ +\end_inset + + +\end_layout + +\begin_layout Standard +Za posplošeno diagonalno matriko +\begin_inset Formula $D$ +\end_inset + + velja +\begin_inset Formula $\forall i,j:i\not=j\Rightarrow D_{ij}=0$ +\end_inset + + +\end_layout + +\begin_layout Standard +Izred o SVD: +\begin_inset Formula $\forall A\in M_{m\times n}\left(\mathbb{C}\right)\exists\text{unit. }Q_{1},\text{unit. }Q_{2},\text{diag. }D\ni:A=Q_{1}DQ_{2}^{-1}=Q_{1}DQ_{2}^{*}$ +\end_inset + +. + Diagonalci +\begin_inset Formula $D$ +\end_inset + + so singvr +\begin_inset Formula $A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $A^{*}=Q_{2}D^{*}Q_{1}^{*}$ +\end_inset + +, +\begin_inset Formula $A^{*}A=Q_{2}D^{*}DQ_{1}^{*}\sim D^{*}D$ +\end_inset + +. + Diagonalci +\begin_inset Formula $D^{*}D$ +\end_inset + + so lavr +\begin_inset Formula $A^{*}A$ +\end_inset + + in stolpci +\begin_inset Formula $Q_{2}$ +\end_inset + + so ONB lave +\begin_inset Formula $A^{*}A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Konstrukcija +\begin_inset Formula $Q_{2}$ +\end_inset + +: ONB iz pripadajočih ONB +\begin_inset Formula $A^{*}A$ +\end_inset + +. + +\begin_inset Formula $r=\text{rang}A$ +\end_inset + + +\end_layout + +\begin_layout Standard +Konstrukcija +\begin_inset Formula $Q_{1}$ +\end_inset + +: +\begin_inset Formula $\forall i\in\left\{ 1..r\right\} :u_{i}=\frac{1}{\sigma_{i}}Av_{i}$ +\end_inset + +. + +\begin_inset Formula $\left\{ u_{1},\dots,u_{r}\right\} $ +\end_inset + + dopolnimo do ONB, +\begin_inset Formula $Q_{1}=\left[\begin{array}{ccccc} +u_{1} & \cdots & u_{r} & \cdots & u_{m}\end{array}\right]$ +\end_inset + + unitarna (ONB stolpci) +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{multicols} +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/šola/p2/dn/DN07a_63230317.c b/šola/p2/dn/DN07a_63230317.c new file mode 100644 index 0000000..3bc5323 --- /dev/null +++ b/šola/p2/dn/DN07a_63230317.c @@ -0,0 +1,25 @@ +#include <stdio.h> +#include <stdlib.h> +int globina (int * t) { + fprintf(stderr, "-> %d %d\n", t[0], t[1]); + if (!t[0] && !t[1]) + return 0; + int r = 0; + if (t[0]) + r = globina(t+2*t[0]); + if (t[1]) { + int g = globina(t+2*t[1]); + if (g > r) + r = g; + } + return r+1; + +} +int main (void) { + int n; + scanf("%d\n", &n); + int t[2*n]; + for (int i = 0; i < 2*n; i++) + scanf("%d", &t[i]); + printf("%d\n", globina(t)); +} diff --git a/šola/p2/dn/DN07b_63230317.c b/šola/p2/dn/DN07b_63230317.c new file mode 100644 index 0000000..72a1ee9 --- /dev/null +++ b/šola/p2/dn/DN07b_63230317.c @@ -0,0 +1,24 @@ +#include <stdio.h> +#include <stdbool.h> +#include <string.h> +int main (void) { + int n = 0; + scanf("%d\n", &n); + char nizi[n][43]; + int offseti[n]; + memset(offseti, 0, n*sizeof offseti[0]); + for (int i = 0; i < n; i++) + gets(nizi[i]); // izziv je v domačih nalogah pisat čim bolj nevarno a vseeno standardno C kodo + while (true) { + for (int i = 0; i < n; i++) + putchar(nizi[i][offseti[i]]); + putchar('\n'); + offseti[n-1]++; + for (int i = n-1; !nizi[i][offseti[i]]; i--) { + offseti[i] = 0; + offseti[i-1]++; + if (!i) + return 0; + } + } +} diff --git a/šola/p2/dn/DN09b_63230317.c b/šola/p2/dn/DN09b_63230317.c new file mode 100644 index 0000000..daca2bf --- /dev/null +++ b/šola/p2/dn/DN09b_63230317.c @@ -0,0 +1,48 @@ +#include <stdio.h> +#include <stdlib.h> +#include <stdbool.h> +int next (bool * s, int l) { + bool hit0 = false; + bool hit1after0 = false; + int end1count = 0; + int end1 = l-1; + for (int i = l-1; i >= 0; i--) { + if (!s[end1]) + end1--; + if (!hit0 && s[i] == 1) + end1count++; + if (!s[i]) + hit0 = true; + if (hit0 && s[i]) + hit1after0 = true; + } + if (end1 == -1) { // prazen vhod, sedaj 1 bit + for (int i = 0; i < l; i++) + s[i] = false; + s[0] = true; + return 1; + } + if (!hit0) + return len+1; // konec + if (!hif1after0) { // inc št enic + for (int i = 0; i < l; i++) { + s[i] = false; + if (i < end1count) + s[i] = true; + } + return end1count; + } +} +int main (void) { + int s, g, m; + scanf("%d", &s); + char * i[s]; + char * r[s]; + for (int i = 0; i < s; i++) + scanf("%ms %ms", &i[s], &r[s]); + scanf("%d", &g); + char * gl[g]; + for (int i = 0; i < s; i++) + scanf("%ms", &gl[s]); + bool samost[s]; +} |