Bela Bollobas

Modern Graph Theory (eBook, PDF)

• Format: PDF

Produktbeschreibung

An in-depth account of graph theory, written for serious students of mathematics and computer science. It reflects the current state of the subject and emphasises connections with other branches of pure mathematics. Recognising that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavour of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory, the book presents a detailed account of newer topics, including Szemerédis Regularity Lemma and its use, Shelahs extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. Moreover, the book contains over 600 well thought-out exercises: although some are straightforward, most are substantial, and some will stretch even the most able reader.

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Produktdetails

• Verlag: Springer New York
• Seitenzahl: 394
• Erscheinungstermin: 1. Dezember 2013
• Englisch
• ISBN-13: 9781461206194
• Artikelnr.: 44185089

Inhaltsangabe

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.

Rezensionen

"...This book is likely to become a classic, and it deserves to be on the shelf of everyone working in graph theory or even remotely related areas, from graduate student to active researcher."--MATHEMATICAL REVIEWS