Book:Béla Bollobás/Modern Graph Theory

Subject Matter

 * Graph Theory

Contents

 * Apologia
 * Preface


 * I Fundamentals
 * I.1 Definitions
 * I.2 Paths, Cycles, and Trees
 * I.3 Hamilton Cycles and Euler Circuits
 * I.4 Planar Graphs
 * I.5 An Application of Euler Trails to Algebra
 * I.6 Exercises


 * II Electrical Networks
 * II.1 Graphs and Electrical Networks
 * II.2 Squaring the Square
 * II.3 Vector Spaces and Matrices Associated with Graphs
 * II.4 Exercises
 * II.5 Notes


 * III Flows, Connectivity and Matching
 * III.1 Flows in Directed Graphs
 * III.2 Connectivity and Menger's Theorem
 * III.3 Matching
 * III.4 Tutte's 1-Factor Theorem
 * III.5 Stable Matchings
 * III.6 Exercises
 * III.7 Notes


 * IV Extremal Problems
 * IV.1 Paths and Cycles
 * IV.2 Complete Subgraphs
 * IV.3 Hamilton Paths and Cycles
 * IV.4 The Structure of Graphs
 * IV.5 Szemerédi's Regularity Lemma
 * IV.6 Simple Applications of Szemerédi's Lemma
 * IV.7 Exercises
 * IV.8 Notes


 * V Colouring
 * V.1 Vertex Colouring
 * V.2 Edge Colouring
 * V.3 Graphs on Surfaces
 * V.4 List Colouring
 * V.5 Perfect Graphs
 * V.6 Exercises
 * V.7 Notes


 * VI Ramsey Theory
 * VI.1 The Fundamental Ramsey Theorems
 * VI.2 Canonical Ramsey Theorems
 * VI.3 Ramsey Theory for Graphs
 * VI.4 Ramsey Theory for Integers
 * VI.5 Subsequences
 * VI.6 Exercises
 * VI.7 Notes


 * VII Random Graphs
 * VII.1 The Basic Models&mdash;The Use of the Expectation
 * VII.2 Simple Properties of Almost All Graphs
 * VII.3 Almost Determined Variables&mdash;The Use of the Variance
 * VII.4 Hamilton Cycles&mdash;The Use of Graph Theoretic Tools
 * VII.5 The Phase Transition
 * VII.6 Exercises
 * VII.7 Notes


 * VIII Graphs, Groups and Matrices
 * VIII.1 Cayley and Schreier Diagrams
 * VIII.2 The Adjacency Matrix and the Laplacian
 * VIII.3 Strongly Regular Graphs
 * VIII.4 Enumeration and Pólya's Theorem
 * VIII.5 Exercises


 * IX Random Walks on Graphs
 * IX.1 Electrical Networks Revisited
 * IX.2 Electrical Networks and Random Walks
 * IX.3 Hitting Times and Commute Times
 * IX.4 Conductance and Rapid Mixing
 * IX.5 Exercises
 * IX.6 Notes


 * X The Tutte Polynomial
 * X.1 Basic Properties of the Tutte Polynomial
 * X.2 The Universal Form of the Tutte Polynomial
 * X.3 The Tutte Polynomial in Statistical Mechanics
 * X.4 Special Values of the Tutte Polynomial
 * X.5 A Spanning Tree Expansion of the Tutte Polynomial
 * X.6 Polynomials of Knots and Links
 * X.7 Exercises
 * X.8 Notes


 * Symbol Index
 * Name Index
 * Subject Index