《Modern Graph Theory》

ⅠFundamentals1

Ⅰ.1 Definitions1

Ⅰ.2 Paths，Cycles，and Trees8

Ⅰ.3 Hamilton Cycles and Euler Circuits14

Ⅰ.4 Planar Graphs20

Ⅰ.5 An Application of Euler Trails to Algebra25

Ⅰ.6 Exercises28

ⅡElectcical Networks39

Ⅱ.1 Graphs and Electrical Networks39

Ⅱ.2 Squaring the Square46

Ⅱ.3 Vector Spaces and Matrices Associated with Graphs51

Ⅱ.4 Exercises58

Ⅱ.5 Notes66

ⅢFlows，Connectivity and Matching67

Ⅲ.1 Flows in Directed Graphs68

Ⅲ.2 Connectivity and Menger’s Theorem73

Ⅲ.3 Matching76

Ⅲ.4 Tutte’s 1-Factor Theorem82

Ⅲ.5 Stable Matchings85

Ⅲ.6 Exercises91

Ⅲ.7 Notes101

ⅣExtremal Problems103

Ⅳ.1 Paths and Cycles104

Ⅳ.2 Complete Subgraphs108

Ⅳ.3 Hamilton Paths and Cycles115

Ⅳ.4 The Structure of Graphs120

Ⅳ.5 Szemeredi’s Regularity Lemma124

Ⅳ.6 Simple Applications of Szemeredi’s Lemma130

Ⅳ.7 Exercises135

Ⅳ.8 Notes142

ⅤColouring145

Ⅴ.1 Vertex Colouring146

Ⅴ.2 Edge Colouring152

Ⅴ.3 Graphs on Surfaces154

Ⅴ.4 List Colouring161

Ⅴ.5 Perfect Graphs165

Ⅴ.6 Exercises170

Ⅴ.7 Notes177

ⅥRamsey Theory181

Ⅵ.1 The Fundamental Ramsey Theorems182

Ⅵ.2 Canonical Ramsey Theorems189

Ⅵ.3 Ramsey Theory For Graphs192

Ⅵ.4 Ramsey Theory for Integers197

Ⅵ.5 Subsequences205

Ⅵ.6 Exercises208

Ⅵ.7 Notes213

Ⅶ Random Graphs215

Ⅶ.1 The Basic Models—The Use of the Expectation216

Ⅶ.2 Simple Properties of Almost All Graphs225

Ⅶ.3 Almost Determined Variables—The Use of the Variance228

Ⅶ.4Hamilton Cycles—The Use of Graph Theoretic Tools236

Ⅶ.5 The Phase Transition240

Ⅶ.6 Exercises246

Ⅶ.7 Notes251

Ⅷ Graphs，Groups and Matrices253

Ⅷ.1 Cayley and Schreier Diagrams254

Ⅷ.2 The Adjacency Matrix and the Laplacian262

Ⅷ.3 Strongly Regular Graphs270

Ⅷ.4 Enumeration and P6lya’s Theorem276

Ⅷ.5 Exercises283

ⅨRandom Walks on Graphs295

Ⅸ.1 Electrical Networks Revisited296

Ⅸ.2 Electrical Networks and Random Walks301

Ⅸ.3 Hitting Times and Commute Times309

Ⅸ.4 Conductance and Rapid Mixing319

Ⅸ.5 Exercises327

Ⅸ.6 Notes333

XThe Tutte Polynomial335

X.1 Basic Properties of the Tutte Polynomial336

X.2 The Universal Form of the Tutte Polynomial340

X.3 The Tutte Polynomial in Statistical Mechanics342

X.4 Special Values of the Tutte Polynomial345

X.5 A Spanning Tree Expansion of the Tutte Polynomial350

X.6 Polynomials of Knots and Links358

X.7 Exercises371

X.8 Notes377

Symbol Index379

Name Index383

Subject Index387