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Topics in Algebraic Graph Theory The rapidly expanding area of algebraic graph theory uses two different branches of algebra to explore various aspects of graph theory: linear algebra (for spectral theory) and group theory (for studying graph symmetry). These areas have links with other areas of mathematics, such as logic and harmonic analysis, and are increasingly being used in such areas as computer networks where symmetry is an important feature. Other books cover portions of this material, but this book is unusual in covering both of these aspects and there are no other books with such a wide scope. This book contains ten expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory, linear algebra and group theory. Each chapter concludes with an extensive list of references. LOWELL W. BEINEKE is Schrey Professor of Mathematics at Indiana University- Purdue University Fort Wayne. His graph theory interests include topological graph theory, line graphs, tournaments, decompositions and vulnerability. With Robin J. Wilson he has edited Selected Topics in Graph Theory (3 volumes), Applications of Graph Theory and Graph Connections.Heiscurrently the Editor of the College Mathematics Journal. ROBIN J. WILSON is Head of the Pure Mathematics Department at the Open University and Gresham Professor of Geometry, London. He has written and edited many books on graph theory and combinatorics and on the history of mathematics, including Introduction to Graph Theory and Four Colours Suffice. His interests include graph coloring, spectral graph theory and the history of graph theory and combinatorics. PETER J. CAMERON, internationally recognized for his substantial contributions to the area, served as academic consultant for this volume. He is a professor of mathematics at Queen Mary, University of London, and his mathematical interests are in permutation groups and their operands (logical, algebraic and combinatorial). i ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS FOUNDING EDITOR GIAN-CARLO ROTA Editorial Board R. Doran, P. Flajolet, M. Ismail, T. Y. Lam, E. Lutwak Volume 102 The titles below, and earlier volumes in the series, are available from booksellers or from Cambridge University Press at www.cambridge.org 50 F. Borceux Handbook of Categorical Algebra I 51 F. Borceux Handbook of Categorical Algebra II 52 F. Borceux Handbook of Categorical Algebra III 53 V. F. Kolchin Random Graphs 54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems 55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics 56 V. N. Sachkov Probabilistic Methods in Discrete Mathematics 57 P. M. Cohn Skew Fields 58 R. Gardner Geometric Tomography 59 G. A. Baker Jr. and P. Graves-Morris Pade Approximants, 2ed 60 J. Krajicek Bounded Arithmetic, Propositional Logic and Complexity Theory 61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics 62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory 63 A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65 K. Engel Sperner Theory 66 D. Cvetkovic, P. Rowlinson, and S. Simic Eigenspaces of Graphs 67 F. Bergeron, G. Labelle, and P. Leroux Combinatorial Species and Tree-Like Structures 68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups 69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2ed 70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry 71 G. E. Andrews, R. Askey, and R. Roy Special Functions 72 R. Ticciati Quantum Field Theory for Mathematicians 73 M. Stern Semimodular Lattices 74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I 75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II 76 A. A. Ivanov Geometry of Sporadic Groups I 77 A. Schinzel Polynomials with Special Regard to Reducibility 78 H. Lenz, T. Beth, and D. Jungnickel Design Theory II, 2ed 79 T. Palmer Banach Algebras and the General Theory of ∗-Algebras II 80 O. Stormark Lie’s Structural Approach to PDE Systems 81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables 82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets 83 C. Foias et al. Navier–Stokes Equations and Turbulence 84 B. Polster and G. Steinke Geometries on Surfaces 85 R. B. Paris and D. Karninski Asymptotics and Mellin–Barnes Integrals 86 R. McEliece The Theory of Information and Coding, 2ed 87 B. Magurn Algebraic Introduction to K-Theory 88 T. Mora Systems of Polynomial Equations I 89 K. Bichteler Stochastic Integration with Jumps 90 M. Lothaire Algebraic Combinatorics on Words 91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II 92 P. McMullen and E. Schulte Abstract Regular Polytopes 93 G. Gierz et al. Continuous Lattices and Domains 94 S. Finch Mathematical Constants 95 Y. Jabri The Mountain Pass Theorem 96 G. Gasper, and M. Rahman Basic Hypergeometric Series, 2ed 97 M. C. Pedicchio and W. Tholen Categorical Foundations 99 T. Mora Solving Polynomial Equations Systems II iii From the Illustrated London News, 15 September 1883 Topics in Algebraic Graph Theory Edited by LOWELL W. BEINEKE Indiana University-Purdue University Fort Wayne ROBIN J. WILSON The Open University Academic Consultant PETER J. CAMERON Queen Mary, University of London v CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521801973 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2005 Reprinted 2007 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Topics in algebraic graph theory / edited by Lowell W. Beineke and Robin J. Wilson, academic consultant Peter J. Cameron. p. cm. – (Encyclopedia of mathematics and its applications) Includes bibliographical references and index. ISBN 0-521-80197-4 1. Graph theory. I. Beineke, Lowell W. II. Wilson, Robin J. III. Series. QA166.T64 2004 515’.5 – dc22 2004045915 ISBN 978-0-521-80197-3 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. Contents Preface page xi Foreword by Peter J. Cameron xiii Introduction 1 LOWELL BEINEKE, ROBIN WILSON AND PETER CAMERON 1. Graph theory 1 2. Linear algebra 10 3. Group theory 19 1 Eigenvalues of graphs 30 MICHAEL DOOB 1. Introduction 30 2. Some examples 31 3. A little matrix theory 33 4. Eigenvalues and walks 34 5. Eigenvalues and labellings of graphs 39 6. Lower bounds for the eigenvalues 43 7. Upper bounds for the eigenvalues 47 8. Other matrices related to graphs 50 9. Cospectral graphs 51 2 Graphs and matrices 56 RICHARD A. BRUALDI and BRYAN L. SHADER 1. Introduction 56 2. Some classical theorems 58 3. Digraphs 61 4. Biclique partitions of graphs 67 vii viii Contents 5. Bipartite graphs 69 6. Permanents 72 7. Converting the permanent into the determinant 75 8. Chordal graphs and perfect Gaussian elimination 79 9. Ranking players in tournaments 82 3 Spectral graph theory 88 DRAGOSCVˇ ETKOVIC´ and PETER ROWLINSON 1. Introduction 88 2. Angles 89 3. Star sets and star partitions 94 4. Star complements 96 5. Exceptional graphs 99 6. Reconstructing the characteristic polynomial 101 7. Non-complete extended p-sums of graphs 104 8. Integral graphs 107 4 Graph Laplacians 113 BOJAN MOHAR 1. Introduction 113 2. The Laplacian of a graph 115 3. Laplace eigenvalues 117 4. Eigenvalues and vertex partitions of graphs 122 5. The max-cut problem and semi-definite programming 125 6. Isoperimetric inequalities 127 7. The travelling salesman problem 129 8. Random walks on graphs 130 5Automorphisms of graphs 137 PETER J. CAMERON 1. Graph automorphisms 137 2. Algorithmic aspects 139 3. Automorphisms of typical graphs 140 4. Permutation groups 141 5. Abstract groups 142 6. Cayley graphs 144 7. Vertex-transitive graphs 145 Contents ix 8. Higher symmetry 148 9. Infinite graphs 149 10. Graph homomorphisms 152 9 Cayley graphs 156 BRIAN ALSPACH 1. Introduction 156 2. Recognition 157 3. Special examples 159 4. Prevalence 160 5. Isomorphism 164 6. Enumeration 167 7. Automorphisms 168 8. Subgraphs 169 9. Hamiltonicity 171 10. Factorization 173 11. Embeddings 174 12. Applications 175 7 Finite symmetric graphs 179 CHERYL E. PRAEGER 1. Introduction 179 2. s-arc transitive graphs 181 3. Group-theoretic constructions 182 4. Quotient graphs and primitivity 186 5. Distance-transitive graphs 187 6. Local characterizations 189 7. Normal quotients 192 8. Finding automorphism groups 196 9. A geometric approach 198 10. Related families of graphs 199 8 Strongly regular graphs 203 PETER J. CAMERON 1. An example 203 2. Regularity conditions 205 3. Parameter conditions 206 4. Geometric graphs 208 5. Eigenvalues and their geometry 212 x Contents 6. Rank 3 graphs 214 7. Related classes of graphs 217 9 Distance-transitive graphs 222 ARJEH M. COHEN 1. Introduction 222 2. Distance-transitivity 223 3. Graphs from groups 226 4. Combinatorial properties 230 5. Imprimitivity 233 6. Bounds 235 7. Finite simple groups 236 8. The first step 238 9.
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