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Apologia
Preface
Fundamentals
Definitions
Paths, Cycles, and Trees
Hamilton Cycles and Euler Circuits
Planar Graphs
An Application of Euler Trails to Algebra
Exercises
Notes
Electrical Networks
Graphs and Electrical Networks
Squaring the Square
Vector Spaces and Matrices Associated with Graphs
Exercises
Notes
Flows, Connectivity and Matching
Flows in Directed Graphs
Connectivity and Menger's Theorem
Matching
Tutte's 1-Factor Theorem
Stable Matchings
Exercises
Notes
Extremal Problems
Paths and Cycles
Complete Subgraphs
Hamilton Paths and Cycles
The Structure of Graphs
Szemerédi's Regularity Lemma
Simple Applications of Szemerédi's Lemma
Exercises
Notes
Colouring
Vertex Colouring
Edge Colouring
Graphs on Surfaces
List Colouring
Perfect Graphs
Exercises
Notes
Ramsey Theory
The Fundamental Ramsey Theorems
Canonical Ramsey Theorems
Ramsey Theory For Graphs
Ramsey Theory for Integers
Subsequences
Exercises
Notes
Random Graphs
The Basic Models—The Use of the Expectation
Simple Properties of Almost All Graphs
Almost Determined Variables—The Use of the Variance
Hamilton Cycles—The Use of Graph Theoretic Tools
The Phase Transition
Exercises
Notes
Graphs, Groups and Matrices
Cayley and Schreier Diagrams
The Adjacency Matrix and the Laplacian
Strongly Regular Graphs
Enumeration and Pólya's Theorem
Exercises
Notes
Random Walks on Graphs
Electrical Networks Revisited
Electrical Networks and Random Walks
Hitting Times and Commute Times
Conductance and Rapid Mixing
Exercises
Notes
The Tutte Polynomial
Basic Properties of the Tutte Polynomial
The Universal Form of the Tutte Polynomial
The Tutte Polynomial in Statistical Mechanics
Special Values of the Tutte Polynomial
A Spanning Tree Expansion of the Tutte Polynomial
Polynomials of Knots and Links
Exercises
Notes
Symbol Index
Name Index
Subject Index