Graduate Texts in Mathematics 207

Editorial Board S. Axler F.W. Gehring K.A. Ribet

Springer Science+Business Media, LLC

Graduate Texts in Mathematics

TAKEUTIIZARING. Introduction to 35 ALEXANDERIWERMER. Several Complex Axiomatic Set Theory. 2nd ed. Variables and Banach Algebras. 3rd ed.

2 OXTOBY. Measure and Category. 2nd ed. 36 KELLEy/NAMIOKA et al. Linear 3 SCHAEFER. Topological Vector Spaces. Topological Spaces.

2nd ed. 37 MONK. Mathematical Logic. 4 HILTON/STAMMBACH. A Course in 38 GRAUERTIFRlTZSCHE. Several Complex

Homological Algebra. 2nd ed. Variables. 5 MAC LANE. Categories for the Working 39 ARVESON. An Invitation to C*-Algebras.

Mathematician. 2nd ed. 40 KEMENY/SNELLIKNAPP. Denumerable 6 HUGHES/PIPER. Projective Planes. Markov Chains. 2nd ed. 7 SERRE. A Course in Arithmetic. 41 ApOSTOL. Modular Functions and 8 TAKEUTIIZARING. Axiomatic Set Theory. Dirichlet Series in Number Theory. 9 HUMPHREYs. Introduction to Lie Algebras 2nd ed.

and Representation Theory. 42 SERRE. Linear Representations of Finite 10 COHEN. A Course in Simple Homotopy Groups.

Theory. 43 GILLMAN/JERISON. Rings of Continuous 11 CONWAY. Functions of One Complex Functions.

Variable I. 2nd ed. 44 KENDIG. Elementary Algebraic Geometry. 12 BEALS. Advanced Mathematical Analysis. 45 LoiNE. Probability Theory I. 4th ed. 13 ANDERSONIFULLER. Rings and Categories 46 LOEVE. Probability Theory II. 4th ed.

of Modules. 2nd ed. 47 MOISE. Geometric Topology in 14 GOLUBITSKy/GUILLEMlN. Stable Mappings Dimensions 2 and 3.

and Their Singularities. 48 SACHS/WU. General Relativity for 15 BERBERIAN. Lectures in Functional Mathematicians.

Analysis and Operator Theory. 49 GRUENBERG/WEIR. Linear Geometry. 16 WINTER. The Structure of Fields. 2nd ed. 17 ROSENBLATT. Random Processes. 2nd ed. 50 EDWARDS. Fermat's Last Theorem. 18 HALMOS. Measure Theory. 51 KLINGENBERG. A Course in Differential 19 HALMOS. A Hilbert Space Problem Book. Geometry.

2nd ed. 52 HARTSHORNE. Algebraic Geometry. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 53 MANIN. A Course in Mathematical Logic. 21 HUMPHREYS. Linear Algebraic Groups. 54 GRAvERIWATKlNS. Combinatorics with 22 BARNES/MACK. An Algebraic Introduction Emphasis on the Theory of Graphs.

to Mathematical Logic. 55 BROWNIPEARCY. Introduction to Operator 23 GREUB. Linear Algebra. 4th ed. Theory I: Elements of Functional 24 HOLMES. Geometric Functional Analysis Analysis.

and Its Applications. 56 MASSEY. Algebraic Topology: An 25 HEWITT/STROMBERG. Real and Abstract Introduction.

Analysis. 57 CROWELL/Fox. Introduction to Knot 26 MANES. Algebraic Theories. Theory. 27 KELLEY. General Topology. 58 KOBLITZ. p-adic Numbers, p-adic 28 ZARlsKiiSAMUEL. Commutative Algebra. Analysis, and Zeta-Functions. 2nd ed.

Vol. I. 59 LANG. Cyclotomic Fields. 29 ZARlsKiiSAMUEL. Commutative Algebra. 60 ARNOLD. Mathematical Methods in

Vol.II. Classical Mechanics. 2nd ed. 30 JACOBSON. Lectures in Abstract Algebra I. 61 WHITEHEAD. Elements of Homotopy

Basic Concepts. Theory. 31 JACOBSON. Lectures in Abstract Algebra 62 KARGAPOLOVIMERLZJAKOV. Fundamentals

II. Linear Algebra. of the Theory of Groups. 32 JACOBSON. Lectures in Abstract Algebra 63 BOLLOBAS. Graph Theory.

III. Theory of Fields and Galois Theory. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed. 33 HIRSCH. Differential Topology. 65 WELLS. Differential Analysis on Complex 34 SPITZER. Principles of Random Walk. Manifolds. 2nd ed.

2nd ed. (continued after index)

Chris Godsil Gordon Royle

Algebraic Graph Theory

With 120 lllustrations

, Springer

Chris Godsil Gordon Royle Department of Combinatorics

and Optimization University of Waterloo Waterloo, Ontario N2L 3Gl Canada cgodsil@math.uwaterloo.ca

Department of Computer Science University of Western Australia Nedlands, Western Australia 6907 Australia gordon@cs.uwa.edu.au

Editorial Board S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA

F.w. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA

Mathematics Subject Classification (2000): 05Cxx, 05Exx

Library of Congress Cataloging-in-Publication Data Godsil, C.D. (Christopher David), 1949-

Algebraic graph theory 1 Chris Godsil, Gordon Royle. p. cm. - (Graduate texts in mathematics; 207)

Includes bibliographical references and index.

K.A. Ribet Mathematics Department University of California

at Berkeley Berkeley, CA 94720-3840 USA

ISBN 978-0-387-95220-8 ISBN 978-1-4613-0163-9 (eBook)

DOI 10.1007/978-1-4613-0163-9 1. Graph theory I. Royle, Gordon. ll. Title. ill. Series.

QAl66 .063 2001 511'.5-dc21 00-053776

Printed on acid-free paper.

© 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2001 Softcover reprint of the hardcover 1 st edition 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by A. Orrantia; manufacturing supervised by Jerome Basma. Electronically imposed from the authors' PostScript files.

98765 4 3 2 1

ISBN 978-0-387-95220-8 SPIN 10793786 (hardcover) SPIN 10791962 (softcover)

To Gillian and Jane

Preface

Many authors begin their preface by confidently describing how their book arose. We started this project so long ago, and our memories are so weak, that we could not do this truthfully. Others begin by stating why they de­cided to write. Thanks to Freud, we know that unconscious reasons can be as important as conscious ones, and so this seems impossible, too. More­over, the real question that should be addressed is why the reader should struggle with this text.

Even that question we cannot fully answer, so instead we offer an ex­planation for our own fascination with this subject. It offers the pleasure of seeing many unexpected and useful connections between two beautiful, and apparently unrelated, parts of mathematics: algebra and graph theory. At its lowest level, this is just the feeling of getting something for nothing. After devoting much thought to a graph-theoretical problem, one suddenly realizes that the question is already answered by some lonely algebraic fact. The canonical example is the use of eigenvalue techniques to prove that cer­tain extremal graphs cannot exist, and to constrain the parameters of those that do. Equally unexpected, and equally welcome, is the realization that some complicated algebraic task reduces to a question in graph theory, for example, the classification of groups with BN pairs becomes the study of generalized polygons.

Although the subject goes back much further, Tutte's work was funda­mental. His famous characterization of graphs with no perfect matchings was proved using Pfaffians; eventually, proofs were found that avoided any reference to algebra, but nonetheless, his original approach has proved fruit­ful in modern work developing parallelizable algorithms for determining the

viii Preface

maximum size of a matching in a graph. He showed that the order of the vertex stabilizer of an arc-transitive cubic graph was at most 48. This is still the most surprising result on the autmomorphism groups of graphs, and it has stimulated a vast amount of work by group theorists interested in deriv­ing analogous bounds for arc-transitive graphs with valency greater than three. Tutte took the chromatic polynomial and gave us back the Tutte polynomial, an important generalization that we now find is related to the surprising developments in knot theory connected to the Jones polynomial.

But Tutte's work is not the only significant source. Hoffman and Sin­gleton's study of the maximal graphs with given valency and diameter led them to what they called Moore graphs. Although they were disappointed in that, despite the name, Moore graphs turned out to be very rare, this was nonetheless the occasion for introducing eigenvalue techniques into the study of graph theory.

Moore graphs and generalized polygons led to the theory of distance­regular graphs, first thoroughly explored by Biggs and his collaborators. Generalized polygons were introduced by Tits in the course of his funda­mental work on finite simple groups. The parameters of finite generalized polygons were determined in a famous paper by Feit and Higman; this can still be viewed as one of the key results in algebraic graph theory. Seidel also played a major role. The details of this story are surprising: His work was actually motivated by the study of geometric problems in general metric spaces. This led him to the study of equidistant sets of points in projective space or, equivalently, the subject of equiangular lines. Extremal sets of equiangular lines led in turn to regular two-graphs and strongly regular graphs. Interest in strongly regular graphs was further stimulated when group theorists used them to construct new finite simple groups.

We make some explanation of the philosophy that has governed our choice of material. Our main aim has been to present and illustrate the main tools and ideas of algebraic graph theory, with an emphasis on cur­rent rather than classical topics. We place a strong emphasis on concrete examples, agreeing entirely with H. Liineburg's admonition that " ... the goal of theory is the mastering of examples." We have made a considerable effort to keep our treatment self-contained.

Our view of algebraic graph theory is inclusive; perhaps some readers will be surprised by the range of topics we have treated-fractional chro­matic number, Voronoi polyhedra, a reasonably complete introduction to matroids, graph drawing-to mention the most unlikely. We also find oc­casion to discuss a large fraction of the topics discussed in standard graph theory texts (vertex and edge connectivity, Hamilton cycles, matchings, and colouring problems, to mention some examples).

We turn to the more concrete task of discussing the contents of this book. To begin, a brief summary: automorphisms and homomorphisms, the adjacency and Laplacian matrix, and the rank polynomial.

Preface ix

In the first part of the book we study the automorphisms and homomor­phisms of graphs, particularly vertex-transitive graphs. We introduce the necessary results on graphs and permutation groups, and take care to de­scribe a number of interesting classes of graphs; it seems silly, for example, to take the trouble to prove that a vertex-transitive graph with valency k has vertex connectivity at least 2(k + 1)/3 if the reader is not already in position to write down some classes of vertex-transitive graphs. In addition to results on the connectivity of vertex-transitive graphs, we also present material on matchings and Hamilton cycles.

There are a number of well-known graphs with comparatively large au­tomorphism groups that arise in a wide range of different settings-in particular, the Petersen graph, the Coxeter graph, Tutte's 8-cage, and the Hoffman-Singleton graph. We treat these famous graphs in some detail. We also study graphs arising from projective planes and symplectic forms over 4-dimensional vector spaces. These are examples of generalized polygons, which can be characterized as bipartite graphs with diameter d and girth 2d. Moore graphs can be defined to be graphs with diameter d and girth 2d + 1. It is natural to consider these two classes in the same place, and we do so.

We complete the first part of the book with a treatment of graph homo­morphisms. We discuss Hedetniemi's conjecture in some detail, and provide an extensive treatment of cores (graphs whose endomorphisms are all au­tomorphisms). We prove that the complement of a perfect graph is perfect, offering a short algebraic argument due to Gasparian. We pay particu­lar attention to the Kneser graphs, which enables us to treat fractional chromatic number and the Erdos-Ko-Rado theorem. We determine the chromatic number of the Kneser graphs (using Borsuk's theorem).

The second part of our book is concerned with matrix theory. Chapter 8 provides a course in linear algebra for graph theorists. This includes an extensive, and perhaps nonstandard, treatment of the rank of a matrix. Fol­lowing this we give a thorough treatment of interlacing, which provides one of the most powerful ways of using eigenvalues to obtain graph-theoretic information. We derive the standard bounds on the size of independent sets, but also give bounds on the maximum number of vertices in a bi­partite induced subgraph. We apply interlacing to establish that certain carbon molecules, known as fullerenes, satisfy a stability criterion. We treat strongly regular graphs and two-graphs. The main novelty here is a careful discussion of the relation between the eigenvalues of the subconstituents of a strongly regular graph and those of the graph itself. We use this to study the strongly regular graphs arising as the point graphs of generalized quadrangles, and characterize the generalized quadrangles with lines of size three.

The least eigenvalue of the adjacency matrix of a line graph is at least -2. We present the beautiful work of Cameron, Goethals, Shult, and Seidel, characterizing the graphs with least eigenvalue at least -2. We follow the

x Preface

original proof, which reduces the problem to determining the generalized quadrangles with lines of size three and also reveals a surprising and close connection with the theory of root systems.

Finally we study the Laplacian matrix of a graph. We consider the re­lation between the second-largest eigenvalue of the Laplacian and various interesting graph parameters, such as edge-connectivity. We offer several viewpoints on the relation between the eigenvectors of a graph and various natural graph embeddings. We give a reasonably complete treatment of the cut and flow spaces of a graph, using chip-firing games to provide a novel approach to some aspects of this subject.

The last three chapters are devoted to the connection between graph theory and knot theory. The most startling aspect of this is the connection between the rank polynomial and the Jones polynomial.

For a graph theorist, the Jones polynomial is a specialization of a straightforward generalization of the rank polynomial of a graph. The rank polynomial is best understood in the context of matroid theory, and conse­quently our treatment of it covers a significant part of matroid theory. We make a determined attempt to establish the importance of this polynomial, offering a fairly complete list of its remarkable applications in graph the­ory (and coding theory). We present a version of Tutte's theory of rotors, which allows us to construct nonisomorphic 3-connected graphs with the same rank polynomial.

After this work on the rank polynomial, it is not difficult to derive the Jones polynomial and show that it is a useful knot invariant. In the last chapter we treat more of the graph theory related to knot diagrams. We characterize Gauss codes and show that certain knot theory operations are just topological manifestations of standard results from graph theory, in particular, the theory of circle graphs.

As already noted, our treatment is generally self-contained. We assume familiarity with permutations, subgroups, and homomorphisms of groups. We use the basics of the theory of symmetric matrices, but in this case we do offer a concise treatment of the machinery. We feel that much of the text is accessible to strong undergraduates. Our own experience is that we can cover about three pages of material per lecture. Thus there is enough here for a number of courses, and we feel this book could even be used for a first course in graph theory.

The exercises range widely in difficulty. Occasionally, the notes to a chapter provide a reference to a paper for a solution to an exercise; it is then usually fair to assume that the exercise is at the difficult end of the spectrum. The references at the end of each chapter are intended to provide contact with the relevant literature, but they are not intended to be complete.

It is more than likely that any readers familiar with algebraic graph theory will find their favourite topics slighted; our consolation is the hope

Preface xi

that no two such readers will be able to agree on where we have sinned the most.

Both authors are human, and therefore strongly driven by the desire to edit, emend, and reorganize anyone else's work. One effect of this is that there are very few places in the text where either of us could, with any real confidence or plausibility, blame the other for the unfortunate and inevitable mistakes that remain. In this matter, as in others, our wives, our friends, and our students have made strenuous attempts to point out, and to eradicate, our deficiencies. Nonetheless, some will still show through, and so we must now throw ourselves on our readers' mercy. We do intend, as an exercise in public self-flagellation, to maintain a webpage listing corrections at http://quoll. uwaterloo . cal agt/.

A number of people have read parts of various versions of this book and offered useful comments and advice as a result. In particular, it is a pleasure to acknowledge the help of the following: Rob Beezer, An­thony Bonato, Dom de Caen, Reinhard Diestel, Michael Doob, Jim Geelen, Tommy Jensen, Bruce Richter.

We finish with a special offer of thanks to Norman Biggs, whose own Al­gebraic Graph Theory is largely responsible for our interest in this subject.

Chris Godsil Gordon Royle

Waterloo Perth

Contents

Preface vii

1 Graphs 1 1.1 Graphs .. . . . 1 1.2 Subgraphs ... 3 1.3 Automorphisms 4 1.4 Homomorphisms . 6 1.5 Circulant Graphs 8 1.6 Johnson Graphs 9 1.7 Line Graphs .. 10 1.8 Planar Graphs . 12 Exercises 16 Notes .. 17 References 18

2 Groups 19 2.1 Permutation Groups 19 2.2 Counting ....... 20 2.3 Asymmetric Graphs . 22 2.4 Orbits on Pairs ... 25 2.5 Primitivity . . . . . . 27 2.6 Primitivity and Connectivity. 29 Exercises 30 Notes ... 32 References . 32

xiv Contents

3 Transitive Graphs 33 3.1 Vertex-Transitive Graphs . 33 3.2 Edge-Transitive Graphs. 35 3.3 Edge Connectivity .. 37 3.4 Vertex Connectivity ... 39 3.5 Matchings ........ 43 3.6 Hamilton Paths and Cycles. 45 3.7 Cayley Graphs. . . . . . . . 47 3.8 Directed Cayley Graphs with No Hamilton Cycles 49 3.9 Retracts ... 51 3.10 Transpositions 52 Exercises 54 Notes · .. 56 References . 57

4 Arc-Transitive Graphs 59 4.1 Arc-Transitive Graphs 59 4.2 Arc Graphs ...... 61 4.3 Cubic Arc-Transitive Graphs . 63 4.4 The Petersen Graph. . . . . 64 4.5 Distance-Transitive Graphs. 66 4.6 The Coxeter Graph 69 4.7 Tutte's 8-Cage . 71 Exercises 74 Notes · ........ 76 References . . . . . . . 76

5 Generalized Polygons and Moore Graphs 77 5.1 Incidence Graphs ....... 78 5.2 Projective Planes ....... 79 5.3 A Family of Projective Planes 80 5.4 Generalized Quadrangles . . . 81 5.5 A Family of Generalized Quadrangles 83 5.6 Generalized Polygons . . . . 84 5.7 Two Generalized Hexagons . . . 88 5.8 Moore Graphs .......... 90 5.9 The Hoffman-Singleton Graph . 92 5.10 Designs. 94 Exercises 97 Notes · .. 100 References . 100

6 Homomorphisms 103 6.1 The Basics 103 6.2 Cores .. 104 6.3 Products. 106

6.4 The Map Graph ..... . 6.5 Counting Homomorphisms 6.6 Products and Colourings . 6.7 Uniquely Colour able Graphs 6.8 Foldings and Covers ... 6.9 Cores with No Triangles .. 6.10 The Andnisfai Graphs ... 6.11 Colouring Andrasfai Graphs 6.12 A Characterization ..... 6.13 Cores of Vertex-Transitive Graphs. 6.14 Cores of Cubic Vertex-Transitive Graphs Exercises Notes ... References .

7 Kneser Graphs 7.1 Fractional Colourings and Cliques 7.2 Fractional Cliques ........ . 7.3 Fractional Chromatic Number .. 7.4 Homomorphisms and Fractional Colourings . 7.5 Duality...... .. . 7.6 Imperfect Graphs .. . 7.7 Cyclic Interval Graphs 7.8 Erdos-Ko-Rado .... 7.9 Homomorphisms of Kneser Graphs 7.10 Induced Homomorphisms ..... . 7.ll The Chromatic Number of the Kneser Graph 7.12 Gale's Theorem ... . 7.13 Welzl's Theorem ........ . 7.14 The Cartesian Product .... . 7.15 Strong Products and Colourings Exercises Notes ..... . References . . . .

8 Matrix Theory 8.1 The Adjacency Matrix ........... . 8.2 The Incidence Matrix . . . . . . . . . . . . . 8.3 The Incidence Matrix of an Oriented Graph 8.4 Symmetric Matrices . . . . . . 8.5 Eigenvectors.......... 8.6 Positive Semidefinite Matrices 8.7 Subharmonic Functions . . . . 8.8 The Perron-Frobenius Theorem 8.9 The Rank of a Symmetric Matrix 8.10 The Binary Rank of the Adjacency Matrix

Contents xv

108 109 llO ll3 ll4 ll6 ll8 ll9 121 123 125 128 132 133

135 135 136 137 138 141 142 145 146 148 149 150 152 153 154 155 156 159 160

163 163 165 167 169 171 173 175 178 179 181

xvi Contents

8.11 The Symplectic Graphs . 183 8.12 Spectral Decomposition. 185 8.13 Rational Functions 187 Exercises 188 Notes · ... 192 References . . 192

9 Inter lacing 193 9.1 Interlacing ................ 193 9.2 Inside and Outside the Petersen Graph 195 9.3 Equitable Partitions. . . . . . 195 9.4 Eigenvalues of Kneser Graphs 199 9.5 More Interlacing . . . 202 9.6 More Applications. . 203 9.7 Bipartite Subgraphs . 206 9.8 Fullerenes ...... 208 9.9 Stability of Fullerenes . 210 Exercises 213 Notes · ............ 215 References . . . . . . . . . . . 216

10 Strongly Regular Graphs 217 10.1 Parameters . . . . . . . 218 10.2 Eigenvalues ...... 219 10.3 Some Characterizations . 221 10.4 Latin Square Graphs .. 223 10.5 Small Strongly Regular Graphs 226 10.6 Local Eigenvalues .... 227 10.7 The Krein Bounds. . . . 231 10.8 Generalized Quadrangles 235 10.9 Lines of Size Three ... 237 10.10 Quasi-Symmetric Designs. 239 10.11 The Witt Design On 23 Points 241 10.12 The Symplectic Graphs. 242 Exercises 244 Notes · .... 246 References . . . 247

11 Two-Graphs 249 11.1 Equiangular Lines . 249 11.2 The Absolute Bound 251 11.3 Tightness ....... 252 11.4 The Relative Bound . 253 11.5 Switching ....... 254 11.6 Regular Two-Graphs 256 11.7 Switching and Strongly Regular Graphs 258

11.8 The Two-Graph on 276 Vertices Exercises Notes ... References .

12 Line Graphs and Eigenvalues 12.1 Generalized Line Graphs 12.2 Star-Closed Sets of Lines . 12.3 Reflections.... ..... 12.4 Indecomposable Star-Closed Sets 12.5 A Generating Set 12.6 The Classification 12.7 Root Systems .. 12.8 Consequences .. 12.9 A Strongly Regular Graph Exercises Notes ... References .

13 The Laplacian of a Graph 13.1 The Laplacian Matrix. 13.2 Trees ......... . 13.3 Representations . . . . 13.4 Energy and Eigenvalues. 13.5 Connectivity....... 13.6 Interlacing ....... . 13.7 Conductance and Cutsets . 13.8 How to Draw a Graph .. 13.9 The Generalized Laplacian 13.10 Multiplicities. 13.11 Embeddings Exercises Notes ... References .

14 Cuts and Flows 14.1 The Cut Space. 14.2 The Flow Space 14.3 Planar Graphs . 14.4 Bases and Ear Decompositions. 14.5 Lattices...... .. . 14.6 Duality ........ . 14.7 Integer Cuts and Flows 14.8 Projections and Duals 14.9 Chip Firing 14.10 Two Bounds ..... .

Contents XVll

260 262 263 263

265 265 266 267 268 270 271 272 274 276 277 278 278

279 279 281 284 287 288 290 292 293 295 298 300 302 305 306

307 308 310 312 313 315 316 317 319 321 323

xviii Contents

14.11 Recurrent States. 14.12 Critical States .. 14.13 The Critical Group 14.14 Voronoi Polyhedra. 14.15 Bicycles ..... . 14.16 The Principal Tripartition Exercises . Notes .......... . References . . . . . . . . .

15 The Rank Polynomial 15.1 Rank Functions 15.2 Matroids ..... . 15.3 Duality....... 15.4 Restriction and Contraction 15.5 Codes ............ . 15.6 The Deletion-Contraction Algorithm 15.7 Bicycles in Binary Codes 15.8 Two Graph Polynomials ...... . 15.9 Rank Polynomial .......... . 15.10 Evaluations of the Rank Polynomial. 15.11 The Weight Enumerator of a Code 15.12 Colourings and Codes. 15.13 Signed Matroids . . . . 15.14 Rotors ........ . 15.15 Submodular Functions Exercises . Notes ... References .

16 Knots 16.1 Knots and Their Projections 16.2 Reidemeister Moves .... . 16.3 Signed Plane Graphs ... . 16.4 Reidemeister moves on graphs 16.5 Reidemeister Invariants. 16.6 The Kauffman Bracket 16.7 The Jones Polynomial 16.8 Connectivity. Exercises . Notes ....... . References . . . . . .

17 Knots and Eulerian Cycles 17.1 Eulerian Partitions and Tours 17.2 The Medial Graph ..... .

325 326 327 329 332 334 336 338 338

341 341 343 344 346 347 349 351 353 355 357 358 359 361 363 366 369 371 372

373 374 376 379 381 383 385 386 388 391 392 392

395 395 398

17.3 Link Components and Bicycles. 17.4 Gauss Codes .... 17.5 Chords and Circles .... . 17.6 Flipping Words ...... . 17.7 Characterizing Gauss Codes 17.8 Bent Tours and Spanning Trees 17.9 Bent Partitions and the Rank Polynomial 17.10 Maps ...... . 17.11 Orient able Maps .... . 17.12 Seifert Circles ..... . 17.13 Seifert Circles and Rank Exercises . Notes ....... . References . . . . . .

Glossary of Symbols

Index

Contents xix

400 403 405 407 408 410 413 414 417 419 420 423 424 425

427

433