Graphs and graph models-graph terminology and special types of graphs-representing graphs and graph isomorphism -connectivity-euler and Hamilton

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1 Prepare by Dr. A.R.VIJAYALAKSHMI

2 Graphs a graph moels-graph termology a specal types of graphs-represetg graphs a graph somorphsm -coectty-euler a Hamlto paths

3 Graph Graph: A graph G = (V, E) cossts of V, a oempty set of ertces, a E, a set of uorere pars of stct elemets of V calle eges. For each ee, e = {u, } where u, V. A urecte graph (ot smple) may cota loops. A ege e s a loop f e = {u, u} for some uv. The ertex-set a the ege-set of G are ofte eote by V(G) a E(G), respectely 3

4 Example: 4 Isolate ertex e e e 4 e 5 e 3 3 V,, E e, e, e e e e 3, 4 3, (, ), e3 (, 3), e4 (, 3) 4, 5 4

5 Loop: A ege of a graph that jos a ertex to tself s calle a loop. Multple eges e 4 e e e e loop Multple eges: If two eges of a graph hae the same e ertces the these eges are calle parallel eges or multple eges Pseuo graph: A graph whch has both loops a multple eges s calle a pseuo graph

6 Mult graph: A graph whch has multple eges s calle a mult graph Smple graph: A graph wthout loops a multple eges s calle a smple graph. It s ot smple. It s a smple graph. Icet: If the ege ertex e k the e k s a e ertex of s sa to be cet some wth 6

7 Ajacet ertces: Two ertces are ajacet f they are the e pots of a ege A e 4 B C e e e 3 D Here A a B are ajacet, A a D are ot ajacet Ajacet eges: Two eges are calle ajacet eges f they are cet wth a commo ertex. e e & e,, e 3 are ajacet eges 7

8 Isolate ertex: No ege cet o t s calle a Isolate ertex Mult graph: A graph whch has multple eges s calle a mult graph Fte graph : A graph whose ertex set a ege set are fte Null graph : the graph whose ertex set a eges are empty A B e 3 e C e e 3 D e e Urecte Graph Drecte Graph 8

9 Drecte Graphs (Dgraphs) A recte graph (graph) G cossts of a set V of ertces a a set E of recte eges. A recte ege s a orere par of elemets of V I graph, recte eges are cate by arrows 9

10 Drecte mult graph urecte graph Mxe Graph Dgraph Uerlyg urecte graph

11 Degree: The egree of ertex a graph G, wrtte as ( ), s the umber of eges cet wth, except that each loop at couts twce The maxmal egree s (G ) The mmum egree s (G ) G F Graph termology A E B D C (C) = (H) = (F) =, A ertex of egree oe s calle a peet ertex. H (A) = 4 (B) = 5, (E) = 4 (D) =, egree zero s calle Isolate ertex

12 The orer of a graph G s ts umber of ertces a the sze of a graph s ts umber of eges Number of eges eg at a ertex s the -egree of the ertex a graph a s eote by ( ) Number of eges orgatg from a ertex s the out-egree of the ertex a graph a s eote by () A ertex wth -egree s a source, a a ertex wth out-egree s a sk. A loop at a ertex cotrbutes to both the egree a outegree of ths ertex

13 HANDSHAKING THEOREM Statemet: Let G = (V, E) be a urecte graph wth e eges. The V eg() e. OR I ay graph, the sum of the egrees of all ertces s equal to twce the umber of eges Proof: Each ege of the graph s cet o two ertces. So each cotrbutes a egree to both of ts e ertces a So whle ag the egree of ertces each ege cotrbutes egree.hece V eg() e. 3

14 I a recte graph HANDSHAKING THEOREM E eg ( e) eg ( e) ee Proof: Let G be a smple graph. If e = (u,) s a recte ege, the e cotrbutes oe outegree to u a egree to. Therefore whle fg the sum of the egrees a the sum of the outegrees of all ertces G, for eery coutg of egree of ay ertex, there s a correspog coutg of outegree of some other ertex. Hece the sum S of the egree of all ertces G s equal to the sum S of the outegree of all ee the ertces G. Sce we kow that, The total egrees of all the ertces G = (umber of eges G) () 4

15 Also the total egree of all the ertces = (the sum of egree of all the ertces G) + (the sum of outegree of all the ertces G) = S + S= S (or) S From () we get S = S = (umber of eges G) S S the umber of eges G. 5

16 Theorem: A urecte graph has a ee umber of ertces of o egree. Statemet: Let G = (V, E) be ay graph wth umber of ertces a e umber of eges. Let be the ertces of o egree a be the ertces of ee egree. To proe : k s ee we kow that ( k ),,..., 3,,..., 3 e,where e s the o. of eges G. m ( ) ( j ) e j k m 6

17 Sce each of (j') are ee, m j ( j ) are ee a e s also ee. k ( ) a ee umber a ee umber k ( ) a ee umber. Theorem: The Maxmum egree of ay ertex a smple graph wth ertces s ( ). 7

18 Theorem: The maxmum umber of eges a smple graph wth ertces s ( ) Proof: Let G be a graph wth ertces say,,..., 3 By Hashakg theorem, we hae ( ) e () where e s the umber of eges G. Maxmum egree of ertex a smple graph s ( ).Hece equato () becomes ( ) + ( ) + + ( ) = e ( ) e (. e ) e ( ) Thus the maxmum umber of eges a smple graph wth ertces s ( ) 8

19 Verfy Hashakg theorem (A) = 3, (B) = 3 (C) = 3, (D) = 4, (E) = By Hashakg theorem, we hae e ) ( 7 ) ( ) ( ) ( ) ( ) ( E D C B A A B C e e D E e 4 e 5 6 e 7 e 9

20 , Ca a smple graph exst wth 5 ertces each of egree 5. Sol. We kow that ( ) e (by Ha shakg theorem) 5 5 = e e 75 whch s ot a teger. Such a graph oes ot exst. If a graph cotas eges, 3 ertces of egree 4 a other each of egree3, F the umber of ertces of the graph. Sol. We kow that ( ) e (by Ha shakg theorem) ( 3) + (3 4) = () 3 + = 4 3 = 4 = The umber of ertces of the graph s.

21 Verfy Hashakg theorem B e A e 5 e e 9 e 6 e 4 e E e 7 e Iegree C e 8 e 3 D Outegree ( A ) ( A ) ( B ) 3 Total umber of eges = ( B ) ( C ) 3 ( C ) 5 ( D ) ( E ) Total egree = Total outegree = Number of eges =. ( D ) 3 ( E )

22 Specal types of graphs Regular graph: A graph G = (V,E) whch all the ertces hae the same egree s calle a regular graph If eery ertex a regular graph has egree the the graph s calle -regular graph Example : A regular graph of egree (or) -regular graph regular graph regular graph

23 Complete graph A smple graph wth ertces s sa to be complete f eery ertces s of egree ( ) a s eote by K or A smple graph whch there s exactly oe ege betwee each par of stct ertces, s calle Complete graph Example: 4 K 4 3 K 5 Note:.A complete graph of ertces wll hae (-)/ eges a graph wll hae (-) eges.a complete graph s always a regular graph, sce complete graph sce complete graph K all the ertces hae the same egree a each ertex s of egree ( )

24 Cycles: The a cycle eges C, 3 Cossts of ertces,,,...,,, 3, Example: C3 C 4 C5 Wheels: The wheel graph W s just a cycle graph wth a extra ertex the mle Example: W W 3 4 4

25 Defto: The -cube, eote by Q, s the graph that has ertces represetg the bt strgs of legth. Two ertces are ajacet f the bt strgs that they represet ffer exactly oe bt posto. Q Q Q 3 5

26 Bpartte Graph If the ertex set V of a smple graph G = (V,E) ca be parttoe to subsets V a V such that eery ege of G coects a ertex V a a ertex V ( so that o ege G coects ether two ertces V or two ertces V ), the G s calle a bpartte graph A A Note: K 3 ot Bpartte B B B3 6

27 Example I: Is C 3 bpartte? 3 No, because there s o way to partto the ertces to two sets so that there are o eges wth both Epots the same set. Example : Is C 6 bpartte? 6 5 Yes, because we ca splay C 6 lke ths:

28 Completely bpartte graph If each ertex of V s coecte wth eery ertex of V by a ege, the G s calle a completely bpartte graph. If V cotas m ertces a V cotas ertces the completely bpartte graph s eote by. K m, K 3,4 8

29 Subgraphs A subgraph of a graph G s a graph H such that: V(H) V(G) a E(H) E(G) Example: H H a a b H, H, H 3 are the subgraphs of G K 4 G c H 3 9

30 The complemet of a graph The complemet of G has all the eges that are mssg G.e. that woul hae to be ae to make the complete graph G G K 6 3

31 self-complemetary A smple graph G s sa to be self-complemetary f G a G are somorphc a b a b c c G G Here G a G are somorphc a hece t s self-complemetary graph. 3

32 Proe that f G s self-complemetary the t has 4 or 4 + ertces. Sol. Let G be a self-complemetary wth P ertces, the G has p ( p ) 4 eges. Further oe of Ether 4/ (4 es p) ) (or) 4 /p - p or p - s o. (4 es p - ) p (.e.) (mo 4) or p (mo 4) p (mo 4) or p (mo 4) (.e.) p = 4 or 4 + ertces. 3

33 If the smple graph G has ertces a e eges, how may eges oes G Sol. We kow that (.e.) E(G G ) hae? ( ) K = ( ) E(G) E(G ) ( ) e E(G ) ( ) E(G ) e ( Hece G has ) eges ( ) If the smple graph G has 4 ertces a 5 eges, how may eges oes G hae? Sol. We kow that G has ( ) e eges. 4(4 ). e G has 5 ege 33

34 Uos: Let G = (V, E ) a G = (V, E ) be two smple graphs wth ertex set VV a ege set E E The uo of G, G s GG G G G G = K 5 Coerse: The coerse of a recte graph G = (V,E) s A recte graph (V,F) Where (u, ) F ff (u,) E It s eote by G a b c a b G c G c c 34

35 Isomorphsm Two graphs G a H are sa to be somorphc to each other, f there exsts a oe-to-oe correspoece betwee the ertex sets a the eges sets of G a H whch preseres ajacecy of the ertces. We say G s somorphc to H, wrtte G H Isomorphc graph No-somorphc graph a b a b a b c c c a b c 35

36 Iarats of somorphc graphs Isomorphc graphs hae the same umber of eges a ertces Isomorphc graphs hae the same ertex sequeces. Note: These ca be use to show two graphs are ot somorphc, but ca ot show that two graphs are somorphc. 36

37 Test whether the followg graphs G a H are somorphc. u u u 5 u u 4 u 8 Graph G u 7 u Graph H Soluto:There are 8 ertces a eges both the graphs G a H respectely. Now, I graph G (u ) (u ) 3 (u3 ) (u4) 3 (u5) (u6) 3 (u7 ) (u8) 3 I graph H () 3 () (3) (4) 3 3 (5) 3 (6) (7) (8) 3 37

38 Thus the two graphs agree wth respect to three arats. But ajacecy ot presere, sce the ertces u,, u3, u5 u7 are ajacet to ertex of egree 3 graph G whereas the correspog ertces, 3, 6, 7 are ot ajacet to ertex of egree 3 graph H. Hece the two graphs G a H are ot somorphc. To eterme whether graphs are somorphc, t wll be easer to coser ther matrx represetatos. 38

39 Represetg Graphs a b a b c c Vertex Ajacet Vertces Ital Vertex Termal Vertces a b, c, a c b a, b a c a, c a, b, c a, b, c 39

40 Ajacecy matrx Let G = (V, E) be a graph, V = a E =m The ajacecy matrx of G wrtte A(G), s the x matrx a j where Example a Smple a j b c,, f otherwse j a b c s a ege a b c Note. Ajacecy matrx s symmetrc, s ot uque, base o labelg ertces. Dagoal elemets are zero, f t s smple of G 3. ( ) o. of s the th row or colum 4

41 Example a Not smple b c a b c a b c 3 3 Example 3 a b c a b c a b c 4. Ajacecy matrx s ot symmetrc a recte graph 4

42 Wrte the ajacecy matrx of the graph G (, ),(, ),(, ),(, ),(, ),(, ),(, ),(, ),(, ), Also raw the graph. Sol. The ajacecy matrx s A Graph 4 3 4

43 Icece matrx Let G = (V, E) be a urecte graph, V = a E =m The xm matrx ( b,j ) where s calle the cece matrx e a e e 5 e 4 b c e 3 b j, whe egee, otherwse a b c j s cet o e e e3 e4 e5 Note:. ( ) o. of s the th row. The sum of the etres a colum of the cece matrx for a urecte graph s f e s ot a loop, f e s a loop 43

44 Draw the graph G whose cece matrx s ge below. A Sol. The graph for the ge cece matrx s ge by e e 3 e 6 e 4 e 7 e 3 5 e

45 Determe whether the graphs G a H are somorphc. u u u 5 u u 4 u Graph G Graph H Soluto: There are 6 ertces a 7 eges both the graphs G a H respectely. Now, I graph G (u ) (u ) 3 (u3 ) (u4 ) 3 (u5) (u6 ) I graph H () () (3 ) 3 (4) (5) 3 (6) 45

46 Thus the two graphs agree wth respect to three arats. Further, ) ( 4 6 or u, ) ( 5 3 or u, ) ( or u, ) ( or u, ) ( 5 or u 6 or u ) ( Hece ajacecy presere. The ajacecy matrces of G a H are A G = u u u u u u u u u u u u, A H = Sce A G = A H, the two graphs G a H are somorphc. 46

47 Test whether the followg graphs G a H are somorphc. PI 5 u u u 4 Isomorphc 3 4 u 3 u 5 P a e 3 5 c b P3 u u u3 u4 u 6 f u Not somorphc a a u 3 4 are are ot ajacet ajacet u

48 Path s a sequece of stct ertces such that two cosecute ertces are ajacet(.e hae a ege betwee them) Example: a,, c, b, e s a path a, b, e,, c, b, e, s ot a path; It s a walk a b c e A walk s sa to be a path, f the eges a ertces are stct A walk s calle a tral f all ts eges are stct The umber of eges a path s ts legth. Path a,, c, b, e s of legth 4 A close Path s calle cycle or crcut. Here a,, c, b, e, a s a cycle 48

49 If a graph has ertces a a ertex u s coecte to a ertex, show that there s a path from u to of legth o more tha. Sol. Let u u, u,... u, m, be a path G from u to of legth m. By efto of the path, the ertces u, u, u,... um a As G cotas oly ertces, t follows that m (. e) m Hece for ertces, there s a path from u to of legth o more tha. 49

50 Coecte graph A graph G s sa to be coecte f there exsts at least oe path betwee ay par of ertces of the graph (G) A graph whch s ot coecte s sa to be scoecte graph (H) a b a b H c e G e H H Coecte compoets: A graph that s ot coecte s. the uo of (or) more coecte subgraphs, each par of whch has o ertex commo. These sjot coecte sub graphs are calle the coecte compoets of the Graph. Example: H s the uo of compoets 5

51 Theorem:A graph G s coecte f a oly f for ay partto of V to subsets V a V there s a ege of G jog a ertex (pot) of V to a ertex of V. Proof:Suppose G s coecte. LetV V V bea partto of V to subsets. Let u V a V G s, say coecte, u there exsts,,,, a u - path. Sce Let be the least poste teger such that V ( such a exsts sce = V ).The V a -, are ajacet. Thus theres a le(aege) jog - V a V. - G, 5

52 To proe the coerse, suppose G s ot coecte. The G cotas at least compoets. Let V eote the set of all ertces of oe compoet a V the remag ertces of G. Clearly V V V s a partto of V a there s o ege jog ay pot of V to ay pot of V, whch s a cotracto.hece G s coecte. 5

53 cut-ertex: A cut-ertex of a graph s a ertex whose eleto whose remoal creases the umber of compoets u G e u e u G u G cut-ege: A cut-ege of a graph s a ege whose remoal creases the umber of compoets G e 53

54 Theorem: A smple graph wth ertces a k compoets ca hae at most eges. ) ( ) ( k k Proof: k...,,, 3 Let be the umber of ertces each of k compoets of the graph G. ) (... G V k 3 The e k.). ( ) (... ) ( ) ( ) (, k k Now k k k Squarg o both ses, we get ) ( ) (... ) ( ) ( k k ) ( ]..... [ k k k ) ( k k k k 54

55 ) ( k k k ) ( ) ( k k k Sce G s smple, the maxmum umber of eges of G ts ) ( compoets s Maxmum umber of eges of G = k k ) ( ) ( k k k k ) ( ) ( ) ( k k ) ( ) ( k k (.e.) Maxmum umber of eges of G ) )( ( k k 55

56 Note: If graphs are somorphc,they wll cota crcuts of same legth,k where k> Theorem: If A s the ajacecy matrx of a graph G,the the umber of fferet paths of legth r from to j where r s a poste teger, equal to the (, j) th etry of r A. A path a graph G s calle a Euler path f t clues eery ege exactly oce. A Eulera crcut (Euler Tour) s a crcut whch traerses through all the eges of the graph exactly oce. A graph wth a Eulera crcut s a Eulera graph 56

57 Example: B E C A D 57

58 Theorem: A coecte multgraph has a Euler Tour (Eulera) f a oly f each of ts ertces has a ee egree. Proof: Let G be Eulera. The G has a Euler crcut C (Euler Tour) wth org ertex as u. Each tme a ertex occurs as a teral ertex of C, the two of the eges cet wth are accoute for egree. For teral ertex V(G), () = ( umber of tmes occur se C ) = ee egree. a sce a Euler crcut C cotas eery ege of G starts a es at u. (u) = + {(umber of tmes u occur se C) } = ee egree Thus G has all the ertces of ee egree 58

59 Coersely, let G be a coecte graph whch s ot hag a Euler crcut, wth all ertces of ee egree a less umber of eges. (.e.) ay graph hag less umber of eges tha G, the t has a Euler crcut. Sce each ertex of G has egree at least two, therefore G cotas a close tral. Let C be a close tral of maxmum possble legth G. If C tself has all the eges of G, the C tself a Euler crcut G. If C oes ot coer all eges of G the remoe all eges of C from G a obta the remag graph G' wth E(G') > a C has all the ertces of ee egree, thus the coecte graph G' also has 59

60 all the ertces of ee egree. Sce E(G') < E(G) therefore G' has a Euler crcut C'. Sce G s coecte, there s a ertex both C a C'. Now jo C a C' a traerse all the eges of C a C' wth a commo ertex, we get CC' s a close tral G a E(CC') > E(C), whch s ot possble for the choce of C G has a Euler crcut. Hece G s Eulera graph 6

61 Theorem: A coecte multgraph has a Euler path f t has exactly two ertces of o egree. Proof: Suppose that G s coecte graph wth two ertces u a of o egree. Coser the graph G + e, (e = u) each ertex of G + e ee egree, the G + e has a Euler crcut. Let C be a Euler crcut, the elete a ege e from C, we get a Euler path C e of G. 6

62 The KÖgsberg Brge Problem Kögsber s a cty o the Pregel rer Prussa The cty occupe two slas plus areas o both baks Problem: Whether they coul leae home, cross eery brge exactly oce, a retur home. Represet the problem by meas of graph. Does the problem hae a soluto?. X W Y Z Fg () 6

63 Soluto: There are slas W &Y forme by rer. They are coecte to each other a to the rer baks X& Z by meas of 7 brges as show Fg () W e e e 3 e 4 e 6 Y e 5 e7 Z Fg () areas x, y,w,z walk across each brge exactly oce a retur to the startg pot. Whe t s represete by a graph, wth ertces represetg the la areas a the eges the brges, the graph wll be show the Fg ().The graph has all the ertces wth o X The problem s to start from ay oe of the 4 la Degree. Hece t has o soluto.we ca t trael all the eges exactly oce by startg at ay oe of la area 63

64 F a Euler path or a Eulera crcut, f t exsts the ge graph. Euler path s No Eulera crcut A B F C Euler path a Euler crcut oes ot exsts E D (sce all the ertces s of o egree) B A C Euler path a Euler crcut exsts E D A B C D E A C E B D A. 64

65 Hamltoa path: A path of a graph G whch clues each ertex of G exactly oce. Hamltoa crcut: A crcut of a graph G whch clues each ertex of G exactly oce, except the startg a e ertces whch appear twce. Hamltoa graph: A graph cotag a Hamltoa crcut s calle a Hamltoa graph Example: B C A D 65

66 F a Hamltoa path or a Hamltoa crcut, f t exsts each A B of the followg graphs. If t oes ot exst, expla why? P A C E B D F Hamltoa crcut amely A B C D E F A, A B C F E D A A B E D C F A, A B E F C D A, A D C B E F A P A B C Nether a Hamltoa path or a Hamltoa crcut sce ay path cotag all the ertces must cota oe of the eges A B a E F more tha oce. D E F A B P3 C D E Hamltoa paths from C to E a from D to E C B D A E a D B C A E But o Hamltoa crcut. 66

67 Strogly coecte graph: A graph s sa to be strogly coecte, f for ay for ay par of oes u a G, there s a path from u to a a path from to u.(.e) each s reachable from the other.(sequece of recte ege s must). Example: A B D C Weakly coecte graph: A graph s sa to be weakly coecte, f there s a path betwee eery ertces the uerlyg urecte graph 67

68 Real lfe examples where graphs are useful Ctes a Roas Networks: Routers are ertces Lks are eges Shortest path betwee ertces. Costrat represetato: If A s there, B ca ot be there, etc. 68

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