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Apologia Preface ¢ñ Fundamentals ¢ñ.1 Definitions ¢ñ.2 Paths, Cycles, and Trees ¢ñ.3 Hamilton Cycles and Euler Circuits ¢ñ.4 Planar Graphs ¢ñ.5 An Application of Euler Trails to Algebra ¢ñ.6 Exercises ¢ò Electrical Networks ¢ò.1 Graphs and Electrical Networks ¢ò.2 Squaring the Square ¢ò.3 Vector Spaces and Matrices Associated with Graphs ¢ò.4 Exercises ¢ò.5 Notes ¢ó Flows, Connect¢ôity and Matching ¢ó.1 Flows in Directed Graphs ¢ó.2 Connect¢ôity and Menger's Theorem ¢ó.3 Matching ¢ó.4 Tutte's 1-Factor Theorem ¢ó.5 Stable Matchings ¢ó.6 Exercises ¢ó.7 Notes ¢ô Extremal Problems ¢ô.1 Paths and Cycles ¢ô.2 Complete Subgraphs ¢ô.3 Hamilton Paths and Cycles ¢ô.4 The Structure of Graphs ¢ô.5 Szemeredi's Regularity Lemma ¢ô.6 Simple Applications of Szemer&ti's Lernma ¢ô.7 Exercises ¢ô.8 Notes ¢õ Colouring ¢õ.1 Vertex Colouring ¢õ.2 Edge Colouring ¢õ.3 Graphs on Surfaces ¢õ.4 List Colouring ¢õ.5 Perfect Graphs ¢õ.6 Exercises ¢õ.7 Notes ¢ö Ramsey Theory ¢ö.1 The Fundamental Rarnsey Theorems ¢ö.2 Canonical Ramsey Theorems ¢ö.3 Ramsey Theory For Graphs ¢ö.4 Ramsey Theory for Integers ¢ö.5 Subsequances ¢ö.6 Exercises ¢ö.7 Notes ¢÷ Random Graphs ¢÷.1 The Basic Models---The Use of the Expectation ¢÷.2 Simple Properties of Almost All Graphs ¢÷.3 Almost Determined VariableskThe Use of the Variance ¢÷.4 Hamilton Cycles---The Use of Graph Theoretic Tools. ¢÷.5 The Phase Transition ¢÷.6 Exercises ¢÷.7 Notes ¢ø Graphs, Groups and Matrices ¢ø.1 Cayley and Schreier Diagrams ¢ø.2 The Adjacency Matrix and the Laplacian ¢ø.3 Strongly Regular Graphs ¢ø.4 Enumeration and P61ya's Theorem ¢ø.5 Exercises ¢ù Random Walks on Graphs ¢ù.1 Electrical Networks Revisited ¢ù.2 Electrical Networks and Random Walks ¢ù.3 Hitting Times and Commute Times ¢ù.4 Conductance and Rapid Mixing ¢ù.5 Exercises ¢ù.6 Notes ¢ú The TaRe Polynomial ¢ú.1 Basic Properties of the Tutte Polynomial ¢ú.2 The Un¢ôersal Form of the Tutte Polynomial ¢ú.3 The Tutte Polynomial in Statistical Mechanics ¢ú.4 Special Values of the "luRe Polynomial ¢ú.5 A Spanning Tree Expansion of the Tutte Polynomial ¢ú.6 Polynomials of Knots and Links ¢ú.7 Exercises ¢ú.8 Notes Symbol Index Name Index Subject Index