Modern Graph Theory

Graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in -depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject.

Zusammenfassung
Presents an account of graph theory. Written with students of mathematics and computer science in mind, this book reflects the state of the subject and emphasizes connections with other branches of pure mathematics. It presents a survey of fresh topics and includes more than 600 exercises.

Inhalt
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.- W.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-The Use of the Expectation.- VII.2 Simple Properties of Almost All Graphs.- VII.3 Almost Determined Variables-The Use of the Variance.- VII.4 Hamilton Cycles-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.2The 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.

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