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Definition 3.2 A connected graph T is a tree if T has no cycles as its subgraphs.¡–DI‚‚€‚ƒ&ª@E  X    ó Ÿ¨Bridges¡‚ª  Ÿ žA component of a graph is a maximal connected subgraph. Let c(G) denote the number of components in G. A graph is connected if and only if c(G) = 1. An edge e of a graph G is a bridge if c(G  e) > c(G). ¡hÐ ‚2‚sƒª&.  š ó Ÿ¨Equivalent Statements¡‚ª  Ÿ JTheorem 3.3 Let T be a graph of order p. Then the following statements are equivalent. T is a tree. T is connected and T has p  1 edges. T has no cycles and T has p  1 edges. Every edge of T is a bridge. 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&ÿÿÿÿrialBlackmanÜ1|ÚdÚì 0|ÚWo 0h"@·DWingdingskmanÜ1|ÚdÚì 0|ÚWo 0hP·DSymbolgskmanÜ1|ÚdÚì 0|ÚWo 0h¤ €`€¥ .©  @£nÿý?" dd@ÿÿï€ÿÿ  @@``€€ lðdðØd$ ð,bð$– hç­�ïà Á6¸ÏÿÀƒ ð0…‡�ƒ¿Àÿ@ñ÷ð8ó€ó€Ð¥ºuìÊš;2NÍÉÊš;ÿ úgþý4CdCdì 0pÚ¬ÿÿÿ¼þÿÿpûppû@ <ý4!d!d® 0,ÞÈ2<ý4dddd® 0,ÞÈ2ˆfŠ^º___PPT9‹@®8¯¬ €ÿÿ ?Ù Ú%ðróŸ¨Trees¡‚<ª  Ÿ¨%Lecture 3 : The most beautiful graphs¡J& ‚,‚,‚,‚,ª & ó Ÿ¨Connected Graphs¡‚ª  Ÿ VDefinition 3.1 A graph G is a connected graph if its diameter diam(G) < +µð. Definition 3.2 A connected graph T is a tree if T has no cycles as its subgraphs.¡–DI‚‚€‚ƒ&ª@E  X    ó Ÿ¨Bridges¡‚ª  Ÿ žA component of a graph is a maximal connected subgraph. Let c(G) denote the number of components in G. A graph is connected if and only if c(G) = 1. An edge e of a graph G is a bridge if c(G  e) > c(G). ¡hÐ ‚2‚sƒª&.  š ó Ÿ¨Equivalent Statements¡‚ª  Ÿ JTheorem 3.3 Let T be a graph of order p. Then the following statements are equivalent. T is a tree. T is connected and T has p  1 edges. T has no cycles and T has p  1 edges. Every edge of T is a bridge. For each pair of vertices u and v in T there exists a unique path from u to v. ¡R ZRZÈ’­Z ‚RÈ€ª & ó Ÿ¨Degree Sequences¡‚ª  Ÿ ÞDefinition 3.4 Let G be a graph with vertex set {v1,v2, & ,vp}. {degG(vi) = di}i = 1 to p is called a degree sequence of G. For convenience, we may assume that d1 ³ð d2 ³ð & ³ð dp. Theorem 3.5 If D = < d1,d2, & ,dp> is a sequence of non-decreasing positive integers such that Sð di = 2p  2, then there exists a tree T of order p whose degree sequence is D. ¡šZ«Z ZªZ‚) çÿ çÿ çÿ çÿ çÿ çÿ çÿ ‚,çÿ†Žçÿ†Žçÿ‚ †‚††‚††‚†‚çÿ†Žçÿ†ŽçÿA†ŽçÿL†Žçÿª¨?      a  &  @  M êîäï € ”ðŒpðdð$ð( ð ðdðr ðd S ð€œ/[¿ÿðP`À0ðà  [ ð žðr ðd S ð€àv[¿ÿ ð�ðÀ�ðà [ ð žðH ðd ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ___3™Ì™ÿ3f3r¶B Yõí’Bk[ ìè½é(€àà€ ò’/È 0ÒÕÈ·DTimes New RomanÜ1|ÚdÚì 0|ÚWo 0h·D°e0}fÔš New RomanÜ1|ÚdÚì 0|ÚWo 0ˆh ·DArial BlackmanÜ1|ÚdÚì 0|ÚWo 0h"0·DArialBlackmanÜ1|ÚdÚì 0|ÚWo 0h"@·DWingdingskmanÜ1|ÚdÚì 0|ÚWo 0hP·DSymbolgskmanÜ1|ÚdÚì 0|ÚWo 0h¤ €`€¥ .©  @£nÿý?" dd@ÿÿï€ÿÿ  @@``€€ tðlðàh' ð,bð$– hç­�ïà Á6¸ÏÿÀƒ ð0…‡�ƒ¿Àÿ@ñ÷ð8ó€ó€Ð¥ºuìÊš;2NÍÉÊš;ÿ úgþý4CdCdì 0pÚ¬ÿÿÿ¼þÿÿpûppû@ <ý4!d!d® 0,ÞÈ2<ý4dddd® 0,ÞÈ2ˆfŠ^º___PPT9‹@®8¯¬ €ÿÿ ?Ù Ú%ðfóŸ¨Trees¡‚<ª  Ÿ¨%Lecture 3 : The most beautiful graphs¡J& ‚,‚,‚,‚,ª & ó Ÿ¨Connected Graphs¡‚ª  Ÿ VDefinition 3.1 A graph G is a connected graph if its   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`þÿÿÿbcdefghijklmnopqrstu´wxyz{|}~€�‚ƒ„…†‡ˆ‰Š‹Œ�Ž��‘’“”•–—˜™š›œ�žŸþÿÿÿ¡¢£¤¥¦§¨©ª«¬­®¯Äýÿÿÿýÿÿÿ³þÿÿÿµ¶·¸¹º»¼½¾¿À þÿÿÿÃþÿÿÿÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖרÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿþÿÿÿRoot Entryÿÿÿÿÿÿÿÿ��d›Oφꪹ)è0¤ÈfUÐÅÂPicturesÿÿÿÿÿÿÿÿÀCurrent Userÿÿÿÿÿÿÿÿÿÿÿÿ ,SummaryInformation(ÿÿÿÿvLRPowerPoint Document(ÿÿÿÿÿÿÿÿÿÿÿÿa­ÚDocumentSummaryInformation8ÿÿÿÿÿÿÿÿŒÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ€ ò’/È 0ÒÕÈ·DTimes New RomanÜ1|ÚdÚì 0|ÚWo 0h·D°e0}fÔš New RomanÜ1|ÚdÚì 0|ÚWo 0ˆh ·DArial BlackmanÜ1|ÚdÚì 0|ÚWo 0h"0·DArialBlackmanÜ1|ÚdÚì 0|ÚWo 0h"@·DWingdingskmanÜ1|ÚdÚì 0|ÚWo 0hP·DSymbolgskmanÜ1|ÚdÚì 0|ÚWo 0h¤ €`€¥ .©  @£nÿý?" dd@ÿÿï€ÿÿ  @@``€€ lðdðØd$ ð,bð$– hç­�ïà Á6¸ÏÿÀƒ ð0…‡�ƒ¿Àÿ@ñ÷ð8ó€ó€Ð¥ºuìÊš;2NÍÉÊš;ÿ úgþý4CdCdì 0pÚ¬ÿÿÿ¼þÿÿpûppû@ <ý4!d!d® 0,ÞÈ2<ý4dddd® 0,ÞÈ2ˆfŠ^º___PPT9‹@®8¯¬ €ÿÿ ðfóŸ¨Trees¡‚<ª  Ÿ¨%Lecture 3 : The most beautiful graphs¡J& ‚,‚,‚,‚,ª & ó Ÿ¨Connected Graphs¡‚ª  Ÿ VDefinition 3.1 A graph G is a connected graph if its diameter diam(G) < +µð. Definition 3.2 A connected graph T is a tree if T has no cycles as its subgraphs.¡–DI‚‚€‚ƒ&ª@E  X    ó Ÿ¨Bridges¡‚ª  Ÿ žA component of a graph is a maximal connected subgraph. Let c(G) denote the number of components in G. A graph is connected if and only if c(G) = 1. An edge e of a graph G is a bridge if c(G  e) > c(G). ¡hÐ ‚2‚sƒª&.  š ó Ÿ¨Equivalent Statements¡‚ª  Ÿ JTheorem 3.3 Let T be a graph of order p. Then the following statements are equivalent. T is a tree. T is connected and T has p  1 edges. T has no cycles and T has p  1 edges. Every edge of T is a bridge. For each pair of vertices u and v in T there exists a unique path from u to v. ¡R ZRZÈ’­Z ‚RÈ€ª & ó Ÿ¨Degree Sequences¡‚ª  Ÿ ôDefinition 3.4 Let G be a graph with vertex set {v1,v2, & ,vp}. {degG(vi) = di}i = 1 to p is called a degree sequence of G. For convenience, we may assume that d1 ³ð d2 ³ð & ³ð dp. Theorem 3.5 Prove that if D = < d1,d2, & ,dp> is a sequence of non-decreasing positive integers such that Sð di = 2p  2, then there exists a tree T of order p whose degree sequence is D. ¡xZ«Z ZµZ‚) çÿ çÿ çÿ çÿ çÿ çÿ çÿ ‚,çÿ†Žçÿ†Žçÿ‚†‚ ††‚†‚çÿ†Žçÿ†ŽçÿA†ŽçÿL†Žçÿª¨?      a  1  @  M êîäï € ”ðŒpðdð$ð( ð ðdðr ðd S ð€œ/[¿ÿðP`À0ðà  [ ð žðr ðd S ð€àv[¿ÿ ð�ðÀ�ðà [ ð žðH ðd ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ___3™Ì™ÿ3f3rå) Ž@õíÁ)zB ìèÁé(€àà€ ò’/È 0ÒÕÈ·DTimes New RomanÜ1|ÚdÚì 0|ÚWo 0h·D°e0}fÔš New RomanÜ1|ÚdÚì 0|ÚWo 0ˆh ·DArial BlackmanÜ1|ÚdÚì 0|ÚWo 0h"0·DA þÿÿÿþÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþÿÕÍÕœ.“—+,ù®0\ˆ�¤´¼ÄÌ Ô Üä ìôü  ¶ ¦p¿Ã¹õ¤j¤pA-Mathj­Ú%í Times New Roman ·s²Ó©úÅé Arial BlackArial WingdingsSymbolBambooTreesConnected GraphsBridgesEquivalent StatementsDegree SequencesSpanning TreesThe Number of Spanning Trees  ¨Ï¥Î¦r«¬ ²³ø³]­p½d¥» §ë¼v¤ù¼ÐÃö_À‘ã‰ÚôÚhlfuhlfuDdiameter diam(G) < +µð. Definition 3.2 A connected graph T is a tree if T has no cycles as its subgraphs.¡–DI‚‚€‚ƒ&ª@E  X    ó Ÿ¨Bridges¡‚ª  Ÿ žA component of a graph is a maximal connected subgraph. Let c(G) denote the number of components in G. A graph is connected if and only if c(G) = 1. An edge e of a graph G is a bridge if c(G  e) > c(G). ¡hÐ ‚2‚sƒª&.  š ó Ÿ¨Equivalent Statements¡‚ª  Ÿ JTheorem 3.3 Let T be a graph of order p. Then the following statements are equivalent. T is a tree. T is connected and T has p  1 edges. T has no cycles and T has p  1 edges. Every edge of T is a bridge. For each pair of vertices u and v in T there exists a unique path from u to v. ¡R ZRZÈ’­Z ‚RÈ€ª & ó Ÿ¨Degree Sequences¡‚ª  Ÿ ÞDefinition 3.4 Let G be a graph with vertex set {v1,v2, & ,vp}. {degG(vi) = di}i = 1 to p is called a degree sequence of G. For convenience, we may assume that d1 ³ð d2 ³ð & ³ð dp. Theorem 3.5 If D = < d1,d2, & ,dp> is a sequence of non-decreasing positive integers such that Sð di = 2p  2, then there exists a tree T of order p whose degree sequence is D. ¡šZ«Z ZªZ‚) çÿ çÿ çÿ çÿ çÿ çÿ çÿ ‚,çÿ†Žçÿ†Žçÿ‚ †‚††‚††‚†‚çÿ†Žçÿ†ŽçÿA†ŽçÿL†Žçÿª¨?      a  &  @  M ó Ÿ¨Spanning Trees¡‚ª  Ÿ †Definition 3.6 A spanning subgraph H of a connected G is a spanning tree if H itself a tree. Proposition 3.7 Every connected graph contains a spanning tree. Proposition 3.8 Let T be a spanning tree of G and e is an edge of T. Then there exists an edge f of G  T such that T  e + f is a spanning tree of G.¡¶ZTZZ4ZZ�Z‚2 ‚‚3‚�ª&    êîäï € ”ðŒ€ðhð$ð( ð ðhðr ðh S ð€P‚š¿ÿðp�ðPðà  š ð žðr ðh S ð€8wš¿ÿ ðà�pðà š ð žðH ðh ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ___3™Ì™ÿ3f3r§[ lvõíƒ[Xx ìè×é(€àà€ ò’/È 0ÒÕÈ·DTimes New Roman$2Ú|ÚdÚì 0|ÚWo 0·D°e0}fÔš New Roman$2Ú|ÚdÚì 0|ÚWo 0ˆh ·DArial Blackman$2Ú|ÚdÚì 0|ÚWo 0h"0·DArialBlackman$2Ú|ÚdÚì 0|ÚWo 0h"@·DWingdingskman$2Ú|ÚdÚì 0|ÚWo 0hP·DSymbolgskman$2Ú|ÚdÚì 0|ÚWo 0h¤ €`€¥ .©  @£nÿý?" dd@ÿÿï€ÿÿ  @@``€€ |ðtðèp* ð,bð$– hç­�ïà Á6¸ÏÿÀóƒ ð0…‡�ƒ¿Àÿ@ñ÷ð8ó€ó€Ð£ºuìÊš;2NÍÉÊš;ÿ úgþý4BdBdì 0pÚ ÿÿÿ ÿÿÿpûppû@ <ý4!d!d® 0,Þ3Ú<ý4dddd® 0,Þ3ÚˆdŠ\º___PPT9‹>®6¯¬ ?Ù Ú%ðzóŸ¨Trees¡‚<ª  Ÿ¨%Lecture 3 : The most beautiful graphs¡J& ‚,‚,‚,‚,ª%  ó Ÿ¨Connected Graphs¡‚ª  Ÿ VDefinition 3.1 A graph G is a connected graph if its diameter diam(G) < +µð. Definition 3.2 A connected graph T is a tree if T has no cycles as its subgraphs.¡–DI‚‚€‚ƒ&ªNE  X     ó Ÿ¨Bridges¡‚ª  Ÿ žA component of a graph is a maximal connected subgraph. Let c(G) denote the number of components in G. A graph is connected if and only if c(G) = 1. An edge e of a graph G is a bridge if c(G  e) > c(G). ¡hÐ ‚2‚sƒª4.  ™  ó Ÿ¨Equivalent Statements¡‚ª  Ÿ JTheorem 3.3 Let T be a graph of order p. Then the following statements are equivalent. T is a tree. T is connected and T has p  1 edges. T has no cycles and T has p  1 edges. Every edge of T is a bridge. For each pair of vertices u and v in T there exists a unique path from u to v. ¡R ZRZÈ’­Z ‚RÈ€ª%  ó Ÿ¨Degree Sequences¡‚ª  Ÿ ÞDefinition 3.4 Let G be a graph with vertex set {v1,v2, & ,vp}. {degG(vi) = di}i = 1 to p is called a degree sequence of G. For convenience, we may assume that d1 ³ð d2 ³ð & ³ð dp. Theorem 3.5 If D = < d1,d2, & ,dp> is a sequence of non-decreasing positive integers such that Sð di = 2p  2, then there exists a tree T of order p whose degree sequence is D. ¡šZ«Z ZªZ‚) çÿ çÿ çÿ çÿ çÿ çÿ çÿ ‚,çÿ†Žçÿ†Žçÿ‚ †‚††‚††‚†‚çÿ†Žçÿ†ŽçÿA†ŽçÿL†Žçÿª¶?      a  &  @  L  ó Ÿ¨Spanning Trees¡‚ª  Ÿ †Definition 3.6 A spanning subgraph H of a connected G is a spanning tree if H itself a tree. Proposition 3.7 Every connected graph contains a spanning tree. Proposition 3.8 Let T be a spanning tree of G and e is an edge of T. Then there exists an edge f of G  T such that T  e + f is a spanning tree of G.¡¶ZTZZ4ZZ�Z‚2 ‚‚3‚�ª4     óŸ¨The Number of Spanning Trees¡‚ª  Ÿ¨tTheorem 3.9 The number of distinct spanning trees in the complete graph of order p is equal to pp-2. Proof.¡L i ‚Z‚ª u êîäï € ”ðŒPðlð$ð( ð ðlðr ðl S ð€ÝÏ¿ÿðÊ�ðPðà  Ï ð žðr ðl S ð€Œ!Ï¿ÿ ðà� ðà Ï ð žðH ðl ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ___3™Ì™ÿ3f3r”xs•õípx_—æè{é(€àà€ ò’/È 0ÒÕÈ·DTimes New Roman$2Ú|ÚdÚì 0|ÚWo 0·D°e0}fÔš New Roman$2Ú|ÚdÚì 0|ÚWo 0ˆh ·DArial Blackman$2Ú|ÚdÚì 0|ÚWo 0h"0·DArialBlackman$2Ú|ÚdÚì 0|ÚWo 0h"@·DWingdingskman$2Ú|ÚdÚì 0|ÚWo 0hP·DSymbolgskman$2Ú|ÚdÚì 0|ÚWo 0h¤ €`€¥ .©  @£nÿý?" dd@ÿÿï€ÿÿ  @@``€€ |ðtðèt* ð,bð$– hç­�ïà Á6¸ÏÿÀóƒ ð0…‡�ƒ¿Àÿ@ñ÷ð8ó€ó€Ð¥ºuìÊš;2NÍÉÊš;ÿ úgþý4BdBdì 0pÚ ÿÿÿ ÿÿÿpûppû@ <ý4!d!d® 0,Þ3Ú<ý4dddd® 0,Þ3ÚˆfŠ^º___PPT9‹@®8¯¬ €ÿÿ ?Ù Ú%ðóŸ¨Trees¡‚<ª  Ÿ¨%Lecture 3 : The most beautiful graphs¡J& ‚,‚,‚,‚,ª & ó Ÿ¨Connected Graphs¡‚ª Ÿ VDefinition 3.1 A graph G is a connected graph if its diameter diam(G) < +µð. Definition 3.2 A connected graph T is a tree if T has no cycles as its subgraphs.¡–DI‚‚€‚ƒ&ªFE  X    ó Ÿ¨Bridges¡‚ª Ÿ žA component of a graph is a maximal connected subgraph. Let c(G) denote the number of components in G. A graph is connected if and only if c(G) = 1. An edge e of a graph G is a bridge if c(G  e) > c(G). ¡hÐ ‚2‚sƒª*.  š ó Ÿ¨Equivalent Statements¡‚ª  Ÿ pTheorem 3.3 Let T be a graph of order p. Then the following statements are equivalent. T is a tree. T is connected and T has p  1 edges. T has no cycles and T has p  1 edges. T is connected and every edge of T is a bridge. For each pair of vertices u and v in T there exists a unique path from u to v. ¡R ZRZÛ’­Z ‚RÛ€ª 9 ó Ÿ¨Degree Sequences¡‚ª Ÿ ÞDefinition 3.4 Let G be a graph with vertex set {v1,v2, & ,vp}. {degG(vi) = di}i = 1 to p is called a degree sequence of G. For convenience, we may assume that d1 ³ð d2 ³ð & ³ð dp. Theorem 3.5 If D = < d1,d2, & ,dp> is a sequence of non-decreasing positive integers such that Sð di = 2p  2, then there exists a tree T of order p whose degree sequence is D. ¡šZ«Z ZªZ‚) çÿ çÿ çÿ çÿ çÿ çÿ çÿ ‚,çÿ†Žçÿ†Žçÿ‚ †‚††‚††‚†‚çÿ†Žçÿ†ŽçÿA†ŽçÿL†Žçÿª¶?      a  &  @  M ó Ÿ¨Spanning Trees¡‚ª Ÿ †Definition 3.6 A spanning subgraph H of a connected G is a spanning tree if H itself a tree. Proposition 3.7 Every connected graph contains a spanning tree. Proposition 3.8 Let T be a spanning tree of G and e is an edge of T. Then there exists an edge f of G  T such that T  e + f is a spanning tree of G.¡¶ZTZZ4ZZ�Z‚2 ‚‚3‚�ª*    óŸ¨The Number of Spanning Trees¡‚ª  Ÿ¨tTheorem 3.9 The number of distinct spanning trees in the complete graph of order p is equal to pp-2. Proof.¡L i ‚Z‚ª u êîúï € ªð¢Pð`ð:ð( ð ð`ðr ð` S ð€ÒÏ¿ÿðp�ðPðà  Ï ð žðˆ ð` S ð€x×Ï¿ÿ ðà� ðà Ï ð"ž¦ø€p`PpðH ð` ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ___3™Ì™ÿ3f3r›— ´õíw— ¶æè!"é(€àà€ ò’/È 0ÒÕÈ·DTimes New Roman ©Ú|ÚdÚì 0|ÚWo 0h·D°e0}fÔš New Roman ©Ú|ÚdÚì 0|ÚWo 0ˆh ·DArial Blackman ©Ú|ÚdÚì 0|ÚWo 0h"0·DArialBlackman ©Ú|ÚdÚì 0|ÚWo 0h"@·DWingdingskman ©Ú|ÚdÚì 0|ÚWo 0hP·DSymbolgskman ©Ú|ÚdÚì 0|ÚWo 0h¤ €`€¥ .©  @£nÿý?" dd@ÿÿï€ÿÿ  @@``€€ |ðtðèt*  ð,bð$– hç­�ïà Á6¸ÏÿÀóƒ ð0…‡�ƒ¿Àÿ@ñ÷ð8ó€ó€Ð¥ºuìÊš;2NÍÉÊš;ÿ úgþý4BdBdì 0pÚ ÿÿÿRÿÿÿpûppû@ <ý4!d!d® 0,Þ ªÚ<ý4dddd® 0,Þ ªÚˆfŠ^º___PPT9‹@®8¯¬ €ÿÿ ?Ù Ú%ðÂóŸ¨Trees¡‚<ª  Ÿ¨%Lecture 3 : The most beautiful graphs¡J& ‚,‚,‚,‚,ª & ó Ÿ¨Connected Graphs¡‚ª  Ÿ VDefinition 3.1 A graph G is a connected graph if its diameter diam(G) < +µð. Definition 3.2 A connected graph T is a tree if T has no cycles as its subgraphs.¡–DI‚‚€‚ƒ&ª@E  X    ó Ÿ¨Bridges¡‚ª  Ÿ žA component of a graph is a maximal connected subgraph. Let c(G) denote the number of components in G. A graph is connected if and only if c(G) = 1. An edge e of a graph G is a bridge if c(G  e) > c(G). ¡hÐ ‚2‚sƒª&.  š ó Ÿ¨Equivalent Statements¡‚ª  Ÿ pTheorem 3.3 Let T be a graph of order p. Then the following statements are equivalent. T is a tree. T is connected and T has p  1 edges. T has no cycles and T has p  1 edges. T is connected and every edge of T is a bridge. For each pair of vertices u and v in T there exists a unique path from u to v. ¡R ZRZÛ’­Z ‚RÛ€ª 9 ó Ÿ¨Degree Sequences¡‚ª  Ÿ ÞDefinition 3.4 Let G be a graph with vertex set {v1,v2, & ,vp}. {degG(vi) = di}i = 1 to p is called a degree sequence of G. For convenience, we may assume that d1 ³ð d2 ³ð & ³ð dp. Theorem 3.5 If D = < d1,d2, & ,dp> is a sequence of non-decreasing positive integers such that Sð di = 2p  2, then there exists a tree T of order p whose degree sequence is D. ¡šZ«Z ZªZ‚) çÿ çÿ çÿ çÿ çÿ çÿ çÿ ‚,çÿ†Žçÿ†Žçÿ‚ †‚††‚††‚†‚çÿ†Žçÿ†ŽçÿA†ŽçÿL†Žçÿª¨?      a  &  @  M ó Ÿ¨Spanning Trees¡‚ª  Ÿ †Definition 3.6 A spanning subgraph H of a connected G is a spanning tree if H itself a tree. Proposition 3.7 Every connected graph contains a spanning tree. Proposition 3.8 Let T be a spanning tree of G and e is an edge of T. Then there exists an edge f of G  T such that T  e + f is a spanning tree of G.¡¶ZTZZ4ZZ�Z‚2 ‚‚3‚�ª&    óŸ¨The Number of Spanning Trees¡‚ª  Ÿ PTheorem 3.9 The number of distinct spanning trees in the complete graph of order p is equal to pp-2. Proof. Let the complete graph of order Kp be defined on [p] = {1,2,3, & ,p}. Then for each spanning tree T of Kp corresponds to a unique sequence <t1,t2, & ,tp-2> where ti is in [p] and the corresponding rule is as follows: Let s1 be the smallest integer which is a vertex of degree one vertices in T and s1 is adjacent to t1 in T. Then delete s1 from T and continue the above step to find s2 and then t2 & . Now, the proof follows. (?)¡Ò ZZ ‚Z ‚! çÿE çÿ$ çÿ çÿ çÿ  çÿ5 ‚‚çÿK‚‚çÿ‚‚çÿ‚‚çÿ,‚‚çÿ ‚‚çÿ!‚ªt–  E  *      êîäï € ”ðŒPðlð$ð( ð ðlðr ðl S ð€„M¡¿ÿðÊ�ðPðà  ¡ ð žðr ðl S ð€,N¡¿ÿ ðà�°ðà ¡ ð žðH ðl ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ___3™Ì™ÿ3f3r\¶…Øõí8¶qÚæ