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This book is intended as an introduction to graph theory Our aim has been to present what we consider to be the basic material together with a wide
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This book is a collection of topics drawn from the second edition of Graph Theory and Its Applications written by the first two authors of this book
[PDF] Graph Theory and Applications
Graph theory started with Euler who was asked to find a A graph can also be represented by its n × m incidence matrix T For an undirected graph T(i
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GRAPH THEORY WITH APPLICATIONS Problem 1 16 Show that no simple graph has all degrees of its vertices are distinct (i e in a degree sequence of a graph
[PDF] Graph Theory with Algorithms and its Applications - X-Files
This book is a comprehensive text on Graph Theory and the subject matter is presented in an organized and systematic manner This book has been balanced between
[PDF] Graph Theory with Applications to Engineering and Computer Science
Because of its inherent simplicity graph theory has a very wide range of applications in engineering in physical social and biological sciences in
(PDF) Graph Theory and its Applications in Computer Science and
16 juil 2021 · Graph can be used in research areas of computer science such as data mining clustering image capturing networking data structure etc This
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18 sept 2020 · Graph theory is becoming increasingly significant as it is applied to other areas of mathematics science and technology
[PDF] Introduction to Graph Theory
In recent years graph theory has established itself as an important mathematical tool in a wide variety of subjects ranging from operational research and
[PDF] GRAPH THEORY WITH APPLICATIONS
This book is intended as an introduction to graph theory Our aim has been to present what we consider to be the basic material together with a wide
Graph Theory and Its Applications third edition - DOKUMENPUB
This book is a collection of topics drawn from the second edition of Graph Theory and Its Applications written by the first two authors of this book
[PDF] Graph Theory and Applications
Graph theory started with Euler who was asked to find a A graph can also be represented by its n × m incidence matrix T For an undirected graph T(i
[PDF] Graph Theory and Its Applications - MIT Mathematics
In this paper we will discuss how problems like Page ranking and finding the shortest paths can be solved by using Graph Theory At its core graph theory
[PDF] Graph Theory with Applications - Dudhnoi College
GRAPH THEORY WITH APPLICATIONS Problem 1 16 Show that no simple graph has all degrees of its vertices are distinct (i e in a degree sequence of a graph
[PDF] Graph Theory with Algorithms and its Applications - X-Files
This book is a comprehensive text on Graph Theory and the subject matter is presented in an organized and systematic manner This book has been balanced between
[PDF] Graph Theory with Applications to Engineering and Computer Science
Because of its inherent simplicity graph theory has a very wide range of applications in engineering in physical social and biological sciences in
(PDF) Graph Theory and its Applications in Computer Science and
16 juil 2021 · Graph can be used in research areas of computer science such as data mining clustering image capturing networking data structure etc This
[PDF] APPLICATIONS OF GRAPH THEORY - KoreaScience
18 sept 2020 · Graph theory is becoming increasingly significant as it is applied to other areas of mathematics science and technology
[PDF] Introduction to Graph Theory
In recent years graph theory has established itself as an important mathematical tool in a wide variety of subjects ranging from operational research and
What is graph theory and its applications?
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines).How is graph theory used in engineering?
The mathematical structures of graph theory are widely applied in computer science, mathematics and engineering to model relationships between objects in sets of objects. Software engineers use graphs to represent communication networks, data organization, computational devices, the flow of computation and more.How graph theory is useful in computer science?
A graph in this context refers to a collection of vertices or nodes and a collection of edges that connect pairs of vertices [1]. Graph theory can be used in research areas of computer science such as data mining, image segmentation, clustering, image capturing, networking etc.- Graph theory is actively used in various areas biochemistry, engineering and computer sciences; that give hints about its use in problem solving. Intelligent Tutoring Systems are one of the favorite tools appearing as software for using in mathematics problem solving applications.
GRAPHTHEORY
WITHAPPLICATIONS
J.A.BondyandU.S.R.Murty
UniversityofWaterloo,
Ontario,Canada'
NORfH-HOLLAND
NewYork•Amsterdam•Oxford
®J.A.BondyandV.S.R.Muny1976
FirstpublishedinGreatBritain1976by
The·MacmillanPressLtd.
FirstpublishedintheU.S.A.1976by
ElseyierSciencePublishingCo.,Inc.
52VanderbiltAvenue,NewYork,N.Y10017
FifthPrinting,1982.
SoleDistributor
intheU.S.A:ElsevierSciencePublishingCo.
.,Inc.Library
ofCongressCataloginginPublicationDataBondy,JohnAdrian.
Graphtheorywith,applications.
Bibliography:p.
Includes
index.QA166.B671979511'.575-29826·
ISBN7 All formorbyanymeans,withoutpermission.Printed
intheUnitedStatesofAmerica·Toourparents
Preface
variety 'applications' gorithms shouldallbeattempted. arelisted. helpful appendixV. Many usChungphaisan
Preface
vii manuscriptandvaluablesuggestions, andtotheubiquitousG.O.M.forhis kindness andconstantencouragement. B. financialsupport.Finally,wewouldlike toexpressourappreciationtoJoanSelwoodfor
artwork..J.A.Bondy
U.S.R.Murty
Contents
Preface
1GRAPHSANDSUBGRAPHS
1.1GraphsandSimpleGraphs.
1.2GraphIsomorphism
1.3TheIncidenceandAdjacencyMatrices
1.4Subgraphs
1.5VertexDegrees_
1.6Pathsan"dConnection
1.7Cycles._
Applications
1.8The"ShortestPathProblem_
1.,9Sperner'sLemma.
2TREES
2.1Trees
2.2CutEdgesandBonds..
2.3CutV'ertices.
2.4Cayley'sFormula.
Applications.
2.5TheCo"nnectorProblem
3CONNECTIVITY
3.1Connectivity.
3.2Block"s"_
4EULERTOURSAN-nHAMILTONCYCLES"
4.1EulerTours_
4.2HamiltonCycles.
Applications
4.3The",ChinesePostmanProblem
4.4TheTravellin,g'SalesDlanProblem
vi 1 4 7 8 10 12 14 15 2125
27
31
32
,36" ' 42'
44.
47
51
53
62
65
Contents
5MATCHINGS
5.1Matchings
5.2MatchingsandCoveringsinBipartiteGraphs
5.3PerfectMatchings.
Applications
5.4ThePersonnelAssignmentProblem'.
5.5TheOptimalAssignmentProblem
. 6EDGECOLOURINGS6.1EdgeChromaticNumber
6.2Vizing'sTheorem.
Applications
TheTimetablingProblem
7INDEPENDENTSETSANDCLIQUES
7.1IndependentSets.
7.2Ramsey's
7.3Turan'sTheorem.
Applications
7.4Schur'sTheorem.
7.5AGeometryProblem.
8VERTEXCOLOU'RINGS
8.1ChromaticNumber
8.2Brooks'Theorem.
8.3Haj6s'·.
8..4Chromatic
8.5GirthandChromaticNumber
Applications
8.6AStorageProblem
9PLANARGRAPHS
IX 7072
76
80
86
91
93
96
·101
·103,
109·112
·113
·117
·122
123125
129
.131
·163
9.1 9.2 9.3 9.4 9.5' 9.6 9.7, 9.8PlaneandPlanarGraphs.135
DualGraphs..139
Euler'sFormula.143
Bridges..145
Kuratowski's
Theorem.151
Nonhamiltonian
PlanarGraphs..160
Applications
APIa.narityAlgorithm.
x10DIRECTEDGRAPHS
10.1DirectedGraphs.
10.2DirectedPaths
10.3DirectedCycles.
Applications
10.4AJobSequencingPr?blem.
10.5DesigninganEfficientC.omputerDrum
10.6MakingaRoadSystemOne-Way
10.7RankingtheParticipantsinaTournament.
11NETWORKS·
11.1Flows.
11.2 Cuts11.3TheMax-FlowMin-CutTheorem
Applications
11.4Menger'sTheorems
11.5FeasibleFlows
12THECYCLESPACEANDBONDSPACE
12.1CirculationsandPotentialDifferences.
12.2TheNumberofSpanningTrees.
Applications
12.3PerfectSquares.
AppendixIHintstoStarredExercises
AppendixIIISomeInterestingGra.phs.
AppendixIVUnsolvedProblems.
AppendixVSuggestionsforFurtherReading.
Glossary
ofSymbols·IndexContents
·171
·173
·176
·179
·181
·182
·185
·191
·194
·196
·203
206·212
218··220
·227
·232
234·246
·254
·257
·261
1GraphsandSubgraphs
1.1GRAPHSANDSIMPLEGRAPHS
. AgraphGisanorderedtriple(V(G),E(G),t/!G)consistingofa '/erticesIiand'v'arecalledtheendsofe.Exarttple1
G=(\l(G),E(O),t/!G)
whereV(G)-={Vt,V2,V3,V4,vs}
E(G)={el,e2'e3,e4,es,e6,e"es}
andt/JCiisdefinedbyExample2
H=(V(H),E(H),t/!H)
whereV(H)={u,v,w,x,y}
E(H)={a,b,C,d,e,f,g,h}
andisdefinedby t/!H(a)=UV,t/!H(b)=UU,t/!H(C)=VW, t/!H(e)=vx,t/!H(f)=wx,t/!H(g)=ux, t/!H(d)=wx t/!H(h)=xy 2 GGraphTheorywithApplications
b h w HFigure1.1.DiagramsofgraphsGandH
isthis representingvertices lines'edges'. 8, V, V2Figure1.2.AnotherdiagramofG
representingavertexwhichis .possible.GraphsandSubgraphs3
immediately1.1.2).
beprovedinchapter9.) otheredgesofGarelinks. u (0) x (b)Figure1.3.Planarandnonplanargraphs
nontrivial. graphs. edgesingraphG.Moreover,whenjust
write,forinstance,4Graph.TheorywithApplications·
Exercises
isindeedplanar.1.1.3ShowthatifGissimple,thenE
1..2.GRAPHISOMORPHISM
andH.6(Vl)=y,6(V2)=x,O(V3)=U,O(V4)=v,8(v's)=w
and >(et)=h, >(es)=e, >(e2)=g, >(e6)=c, =b, >(e7)=d, >(e4)=a >(es)=f atoneedgejoinsanypairofvertices.) graphonnvertices;itisdenotedbyK n•AdrawingofK
s isshowninfigureGraphsandSubgraphs
(0)(b) 5 (c)Figure1.4.(a)K
5; (b)thecube;(c)K3•3
Exercises
and2differentfromtheonegiven. 1.2.2 vertices. onlyif6(u)6(v)EE(H). 6 1.2.6GraphTheorywithApplications
Showthatthefollowinggraphsareisomorphic:
1.2.7 1.2.81.2.10
1.2.11
1.2.12
Showthat
(a)e(Km,n)=mn; (b) ifGissimpleandbipartite,thenE<:v 2 /4. {n/m}verticesisdenotedbyT m•n•Showthat
e(Tm,n),withequalityonlyifG -Tm,n. O's3-cube.)Showthatthek-cubehas2
k vertices,k2 k-1 edgesandis bipartite. withvertexsetV,twoverticesbeingadjacentinGCifandonly
G isself-complementary,thenv=0,1(mod4). itself. servesadjacency,andthat thesetofsuchpermutationsformaGraphsandSubgraphs7
operationofcomposition. (b)Findf(K n) andf(Km,n). theidentity. vertexset {I,2,3}suchthatf(G)=A. shown morphismgroupofsomegraph.) V2,there.isan
1.3THEINCIDENCEANDADJACENCYMATRICES
v andtheedgesby e.,e2,· · ·,eE•
graph,its e1 e 1 e 2 e 3 e 4e s e 6 e, VIV2V 3v. V11100101VI0211
V21110000V22010
V 30011001
V31 101
V400 01120V
4 10-11M(G)A(G)
V484V3
GFigure1.5
8GraphTheorywithApplications
computers.Exercises
graphG. (a)ShowthateverycolumnsumofMis2.' (b)WhatarethecolumnsumsofA? sothattheadjacencymatrixofGhastheform whereA 21isthetransposeof'A 12 theautomorphismgroupofGisabelian
1.4SUBGRAPHS
, AgraphHisasubgraphofG(writtenHeG)ifV(H)cV(G),E(H)c isa asubgrap-h (orsupergraph)HwithV(H)=V(G),. .underly.ingsimplegraph. xo----ow c x0 cGraphsandSubgraphs9
u - u f yvyvyv 99dbd xwxwx cc
GAspanningG-{u,w}
subgraphofG uuu yvyv gG-{a,b,f}Theinduced
subgraphG[{u,v,x}]
Theedge-induced
5ubgraph
G[{a,C,e,g}]
Figure1.7
subgraphG[V\V']isdenotedbyG -V';itisthesubgraphobtainedfromGV'={v}wewriteG-vforG-{v}.
,E\E'iswrittensimplyasG -E';itisthesubgraphobtainedfromGby setofedgesE'·isdenotedby-G+IfE'={e}wewriteG -eandG+e insteadof0-{e}andG+{e}.LetG1andG
2 besubgraphsofG.WesaythatG t andG 2 aredisjointif common.TheunionG t UG 2 ofG t andG 2 isthe-subgraphwithvertexset10GraphTheorywithApplications
V(GI)UV(Gz)andedgesetE(GI)UE(G
z );ifG 1 andG z aredisjoint,we sometimesdenotetheirunionbyGt+G2•
TheintersectionG
1 nG 2 ofG 1 andG2isdefinedsimilarly,butinthiscaseG 1 andG 2 musthaveatleastone vertexincommon.Exercises
subgraphofK n•1.4.2Showthat
1.4.3DescribehowM(G -E')andM(G -V')canbeobtainedfrom
M(G),andhowA(G -V')canbeobtainedfromA(G).
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