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ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J. L. RODRIGO & W. SADOWSKI (eds) 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S. V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, etale´ homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCIA-PRADA,´ P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAT¸ A˘ & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for graph eigenvalues, Z. STANIC´ 424 Surveys in combinatorics 2015, A. CZUMAJ et al. (eds) 425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) 426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) 427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) 428 Geometry in a Frechet´ context, C.T.J. DODSON, G. GALANIS & E. VASSILIOU 429 Sheaves and functions modulo p, L. TAELMAN 430 Recent progress in the theory of the Euler and NavierStokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LOPEZ´ (eds) 431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL 432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO 433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA 434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA 435 Graded rings and graded Grothendieck groups, R. HAZRAT London Mathematical Society Lecture Note Series: 432
Topics in Graph Automorphisms and Reconstruction
Second Edition
JOSEF LAURI University of Malta
RAFFAELE SCAPELLATO Politecnico di Milano University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence.
www.cambridge.org Information on this title: www.cambridge.org/9781316610442 c Josef Lauri and Raffaele Scapellato 2016 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 2016 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. Library of Congress Cataloguing-in-Publication Data Names: Lauri, Josef, 1955– | Scapellato, Raffaele, 1955– Title: Topics in graph automorphisms and reconstruction / Josef Lauri and Raffaele Scapellato. Description: 2nd edition. | Cambridge : Cambridge University Press, 2016. | Series: London Mathematical Society lecture note series; 432 | Includes bibliographical references and index. Identifiers: LCCN 2016014849 | ISBN 9781316610442 (pbk. : alk. paper) Subjects: LCSH: Graph theory. | Automorphisms. | Reconstruction (Graph theory) Classification: LCC QA166.L39 2016 | DDC 511/.5–dc23 LC record available at https://lccn.loc.gov/2016014849 ISBN 978-1-316-61044-2 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Lil Mary Anne, Christina, Beppe u Sandrina
A Fiorella London Mathematical Society Lecture Note Series: 432
Topics in Graph Automorphisms and Reconstruction
JOSEF LAURI University of Malta
RAFFAELE SCAPELLATO Politecnico di Milano
Contents
Preface to the Second Edition page xi Preface to the First Edition xiii
1 Graphs and Groups: Preliminaries 1 1.1 Graphs and digraphs 1 1.2 Groups 3 1.3 Graphs and groups 7 1.4 Edge-automorphisms and line-graphs 10 1.5 A word on issues of computational complexity 13 1.6 Exercises 15 1.7 Notes and guide to references 17 2 Various Types of Graph Symmetry 18 2.1 Transitivity 18 2.2 Asymmetric graphs 25 2.3 Graph symmetries and the spectrum 29 2.4 Simple eigenvalues 31 2.5 Higher symmetry conditions 32 2.6 Exercises 35 2.7 Notes and guide to references 38 3 Cayley Graphs 39 3.1 Cayley colour graphs 39 3.2 Frucht’s and Bouwer’s Theorems 42 3.3 Cayley graphs and digraphs 44 3.4 The Doyle-Holt Graph 47 3.5 Non-Cayley vertex-transitive graphs 48 3.6 Coset graphs and Sabidussi’s Theorem 49 3.7 Double coset graphs and semisymmetric graphs 51
vii viii Contents
3.8 Hamiltonicity 53 3.9 Characters of abelian groups and Cayley graphs 55 3.10 Growth rates 56 3.11 Exercises 58 3.12 Notes and guide to references 62 4 Orbital Graphs and Strongly Regular Graphs 64 4.1 Definitions and basic properties 64 4.2 Rank 3 groups 68 4.3 Strongly regular graphs 69 4.4 The Integrality Condition 70 4.5 Moore graphs 73 4.6 Exercises 75 4.7 Notes and guide to references 77 5 Graphical Regular Representations and Pseudosimilarity 79 5.1 Elementary results 79 5.2 Abelian groups 80 5.3 Pseudosimilarity 81 5.4 Some basic results 82 5.5 Several pairs of pseudosimilar vertices 84 5.6 Several pairs of pseudosimilar edges 85 5.7 Large sets of mutually pseudosimilar vertices 86 5.8 Exercises 88 5.9 Notes and guide to references 91 6 Products of Graphs 92 6.1 General products of graphs 93 6.2 Direct product 95 6.3 Cartesian product 97 6.4 More products 99 6.5 Stability and two-fold automorphisms 102 6.6 Additional remarks on graph products 105 6.7 Exercises 105 6.8 Notes and guide to references 108 7 Special Classes of Vertex-Transitive Graphs and Digraphs 109 7.1 Generalised Petersen graphs 110 7.2 Kneser graphs and odd graphs 114 7.3 Metacirculant graphs 115 7.4 The quasi-Cayley graphs and digraphs 117 7.5 Generalised Cayley graphs 119 Contents ix
7.6 Exercises 120 7.7 Notes and guide to references 122 8 The Reconstruction Conjectures 123 8.1 Definitions 124 8.2 Some basic results 126 8.3 Maximal planar graphs 132 8.4 Digraphs and degree-associated reconstruction 136 8.5 Exercises 138 8.6 Notes and guide to references 139 9 Reconstructing from Subdecks 140 9.1 The endvertex-deck 140 9.2 Reconstruction numbers 141 9.3 The characteristic polynomial deck 144 9.4 Exercises 147 9.5 Notes and guide to references 147 10 Counting Arguments in Vertex-Reconstruction 149 10.1 Kocay’s Lemma 149 10.2 Counting spanning subgraphs 151 10.3 The characteristic and the chromatic polynomials 154 10.4 Exercises 155 10.5 Notes and guide to references 156 11 Counting Arguments in Edge-Reconstruction 157 11.1 Definitions and notation 157 11.2 Homomorphisms of structures 159 11.3 Lovasz’´ and Nash-Williams’ Theorems 163 11.4 Extensions 166 11.5 Exercises 168 11.6 Notes and guide to references 169 References 171 List of Notations 185 Index of Terms and Definitions 187
Preface to the Second Edition
In this second edition of our book we have tried to maintain the same structure as the first edition, namely a text which, although not providing an exhaustive coverage of graph symmetries and reconstruction, provides a detailed cover- age of some particular areas (generally motivated by our own research inter- est), which is not a haphazard collection of results but which presents a clear pathway through this thick forest. And our aim remains that of producing a text which can relatively quickly guide the reader to the point of being able to understand and carry out research in the topics which we cover. Among the additions in this edition we point out the use of the free com- puter programs GAP, GRAPE and Sage to construct and investigate some well- known graphs, including examples with properties like being semisymmetric, a topic which was treated in the first edition but for which examples are not easy to construct ‘by hand’. We have also updated some chapters with new results, improved the presentation and proofs of others, and introduced short treatments of topics such as character theory of abelian groups and their Cay- ley graphs to emphasise the connection between graph theory and other areas of mathematics. We have corrected a number of errors which we found in the first edition, and for this we would like to thank colleagues who have pointed out several of them, particularly Bill Kocay, Virgilio Pannone and Alex Scott. A special thanks goes to Russell Mizzi for help with overhauling Chapter 6, where we also introduce the new idea of two-fold isomorphisms, and to Leonard Soicher and Matan Zif-Av for several helpful tips regarding the use of GAP and GRAPE. The second author would like to thank the Politecnico di Milano for giving him the opportunity, by means of a sabbatical, to focus on the work needed
xi xii Preface to the Second Edition to complete the current edition of this book. He also thanks the University of Malta for its kind hospitality during this sabbatical. The authors will maintain a list of corrections and addenda at http://staff.um .edu.mt/josef.lauri.
Josef Lauri Raffaele Scapellato Preface to the First Edition
This book arose out of lectures given by the first author to Masters students at the University of Malta and by the second author at the Universita` Cattolica di Brescia. This book is not intended to be an exhaustive coverage of graph theory. There are many excellent texts that do this, some of which are mentioned in the References. Rather, the intention is to provide the reader with a more in-depth coverage of some particular areas of graph theory. The choice of these areas has been largely governed by the research interests of the authors, and the flavour of the topics covered is predominantly algebraic, with emphasis on symmetry properties of graphs. Thus, standard topics such as the automorphism group of a graph, Frucht’s Theorem, Cayley graphs and coset graphs, and orbital graphs are presented early on because they provide the background for most of the work presented in later chapters. Here, more specialised topics are tackled, such as graphical regular representations, pseudosimilarity, graph products, Hamiltonicity of Cayley graphs and special types of vertex-transitive graphs, including a brief treatment of the difficult topic of classifying vertex-transitive graphs. The last four chapters are devoted to the Reconstruction Problem, and even here greater emphasis is given to those results that are of a more algebraic character and involve the symmetry of graphs. A special chapter is devoted to graph products. Such operations are often used to provide new examples from existing ones but are seldom studied for their intrinsic value. Throughout we have tried to present results and proofs, many of which are not usually found in textbooks but have to be looked for in journal papers. Also, we have tried, where possible, to give a treatment of some of these topics that is different from the standard published material (for example, the chapter on graph products and much of the work on reconstruction).
xiii xiv Preface to the First Edition
Although the prerequisites for reading this book are quite modest (exposure to a first course in graph theory and some discrete mathematics, and elemen- tary knowledge about permutation groups and some linear algebra), it was our intention when preparing this book that a student who has mastered its con- tents would be in a good position to understand the current state of research in most of the specialised topics covered, would be able to read with profit journal papers in these areas, and would hopefully have his or her interest sufficiently aroused to consider carrying out research in one of these areas of graph theory. We would finally like to thank Professor Caroline Series for showing an interest in this book when it was still in an early draft form and the staff at Cambridge University Press for their help and encouragement, especially Roger Astley, Senior Editor, Mathematical Sciences, and, for technical help with LATEX, Alison Woollatt, who, with a short style file, solved problems that would have baffled us for ages. Thanks are also due to Elise Oranges, who edited this book thoroughly and pointed out several corrections. The first author would also like to thank the Academic Work Resources Fund Committee and the Computing Services Centre of the University of Malta, the first for some financial help while writing this book and the second for technical assistance. He also thanks his M.Sc. students at the University of Malta, who worked through draft chapters of this book and whose comments and criticism helped to improve the final product.
Josef Lauri Raffaele Scapellato 1 Graphs and Groups: Preliminaries
1.1 Graphs and digraphs In these chapters a graph G = (V(G), E(G)) will consist of two disjoint sets: a nonempty set V = V(G) whose elements will be called vertices and a set E = E(G) whose elements, called edges, will be unordered pairs of distinct elements of V. Unless explicitly stated otherwise, the set of vertices will always be finite. An edge, {u, v}, u, v ∈ V, is also denoted by uv. Sometimes E is allowed to be a multiset, that is, the same edge can be repeated more than once in E. Such edges are called multiple edges. Also, edges uu consisting of a pair of repeated vertices are sometimes allowed; such edges are called loops.But unless otherwise stated, it will always be assumed that a graph does not have loops or multiple edges. The complement of the graph G, denoted by G, has the same vertex-set as G, but two distinct vertices are adjacent in the complement if and only if they are not adjacent in G. The degree of a vertex v, denoted by deg(v), is the number of edges in E(G) to which v belongs. A vertex of degree k is sometimes said to be a k-vertex. Two vertices belonging to the same edge are said to be adjacent, while a vertex and an edge to which it belongs are said to be incident. A loop incident to a vertex v contributes a value of 2 to deg(v). A graph is said to be regular if all of its vertices have the same degree. A regular graph with degree equal to 3 is sometimes called cubic. The minimum and maximum degrees of G are denoted by δ = δ(G) and = (G), respectively. In general, given any two sets A, B, then A−B will denote their set-theoretical difference, that is, the set consisting of all of the elements that are in A but not in B. Also, a set containing k elements is often said to be a k-set. If S is a set of vertices of a graph G, then G−S will denote the graph obtained by removing S from V(G) and removing from E(G) all edges incident to some vertex in S.IfF is a set of edges of G, then G − F will denote the graph whose
1 2 Graphs and Groups: Preliminaries vertex-set is V(G) and whose edge-set is E(G) − F.IfS ={u} and F ={e}, we shall, for short, denote G − S and G − F by G − u and G − e, respectively. If S is a subset of the vertices of G, then G[S] will denote the subgraph of G induced by S, that is, the subgraph consisting of the vertices in S and all of the edges joining pairs of vertices from S. An important modification of the foregoing definition of a graph gives what is called a directed graph,ordigraph for short. In a digraph D = (V(D), A(D)) the set A = A(D) consists of ordered pairs of vertices from V = V(D) and its elements are called arcs.Again,anarc(u, v) is sometimes denoted by uv when it is clear from the context whether we are referring to an arc or an edge. The arc uv is said to be incident to v and incident from u; the vertex u is said to be adjacent to v whereas v is adjacent from u. The number of arcs incident from a ( ) vertex v is called its out-degree, denoted by degout v , while the number of arcs incident to v is called its in-degree and is denoted by degin(v). A digraph is said to be regular if all of its vertices have the same out-degree or, equivalently, the same in-degree. Sometimes, when we need to emphasise the fact that a graph is not directed, we say that it is undirected. The number of vertices of a graph G or digraph D is called its order and is generally denoted by n = n(G) or n = n(D), while the number of edges or arcs is called its size and is denoted by m = m(G) or m = m(D). A sequence of distinct vertices of a graph, v1, v2, ..., vk+1, and edges e1, e2, ..., ek such that each edge ei = vivi+1 is called a path. If we allow v1 and vk+1, and only those, to be the same vertex, then we get what is called a cycle. The length of a path or a cycle in G is the number of edges in the path or cycle. A path of length k is denoted by Pk+1 while a cycle of length k is denoted by Ck.Thedistance between two vertices u, v in a connected graph G, denoted by d(u, v), is the length of the shortest path joining u and v.Thediameter of G is the maximum value attained by d(u, v) as u, v run over V(G), and the girth is the length of the shortest cycle. In these definitions, if we are dealing with a digraph and the ei = vivi+1 are arcs, then the path or cycle is called a directed path or directed cycle, respectively. Given a digraph D,theunderlying graph of D is the graph obtained from ( ) D by considering←→ each pair in A D to be an unordered pair. Given a graph G, the digraph G is obtained from G by replacing each edge in E(G) by a pair of oppositely directed arcs. This way, a graph can always be seen as a special case of a digraph. We adopt the usual convention of representing graphs and digraphs by draw- ings in which each vertex is shown by a dot, each edge by a curve joining the 1.2 Groups 3 corresponding pair of dots and each arc (u, v) by a curve with an arrowhead pointing in the direction from u to v. A number of definitions on graphs and digraphs will be given as they are required. However, several standard graph theoretic terms will be used but not defined in these chapters; these can be found in any of the references [257]or [259].
1.2 Groups A permutation group will be a pair (, Y) where Y is a finite set and is a subgroup of the symmetric group SY , that is, the group of all permutations of Y. The stabiliser of an element y ∈ Y under the action of is denoted by y while the orbit of y is denoted by (y).TheOrbit-Stabiliser Theorem states that, for any element y ∈ Y,
||=|(y)|·|y|. If the elements of Y are all in one orbit, then (, Y) is said to be a transitive permutation group and is said to act transitively on Y. The permutation group is said to act regularly on Y if it acts transitively and the stabiliser of any element of Y is trivial. By the Orbit-Stabiliser Theorem, this is equivalent to saying that acts transitively on Y and ||=|Y|.Also, acts regularly on Y is equivalent to saying that, for any y1, y2 ∈ Y, there exists exactly one α ∈ such that α(y1) = y2. One important regular action of a permutation group arises as follows. Let be any group, let Y = and, for any α ∈ ,letλα be the permutation of Y defined by λα(β) = αβ.LetL() be the set of all permutations λα for all α ∈ . Then (L(), Y) defines a permutation group acting regularly on Y. This is called the left regular representation of the group on itself. One can similarly consider the right regular representation of the group on itself, and this is denoted by (R(), Y). The following is an important generalisation of the previous definitions. If is a group and H ≤ ,letY = /H be the set of left cosets of H in . For any H H H α ∈ ,letλα be a permutation on Y defined by λα (βH) = αβH.LetL () H H be the set of all λα for all α ∈ . Then (L (), Y) defines a permutation group that reduces to the left regular representation of if H ={1}. Two permutation groups (1, Y1), (2, Y2) are said to be equivalent, denoted by (1, Y1) ≡ (2, Y2), if there exists a bijective isomorphism φ : 1 → 2 and a bijection f : Y1 → Y2 such that, for all y ∈ Y1 and for all α ∈ 1, f (α(x)) = φ(α)(f (x)). 4 Graphs and Groups: Preliminaries
Figure 1.1. Aut(G),Aut(H) are isomorphic but not equivalent
In this case we also say that the action of 1 on Y1 is equivalent to the action of 2 on Y2, and sometimes we denote this simply by 1 = 2, when the two sets on which the groups are acting is clear from the context. Figure 1.1 shows a simple example of two graphs whose automorphism groups (to be defined later in this chapter) are isomorphic as abstract groups but clearly not equivalent as permutation groups since the sets (of vertices) on which they act are not equal. (See also Exercise 1.7.) Note in particular that, if (1, Y1) ≡ (2, Y2), then apart from 1 2 as abstract groups, and |Y1|=|Y2|, the cycle structure of the permutations of 1 on Y1 must be the same as those of 2 on Y2. However, the converse is not true; that is, 1 and 2 could be isomorphic and the cycle structures of their respective actions could be the same, but (1, Y1) might not be equivalent to (2, Y2) (see Exercise 1.9). If (, Y) is a permutation group acting on Y and Y is a union of orbits of Y, then we can talk about the action of restricted to Y , that is, the permutation group (, Y ) where, for α ∈ and y ∈ Y , α(y ) is the same as in (, Y). When Y is a union of orbits we also say that it is invariant under the action of because in this case α(y ) ∈ Y for all α ∈ and y ∈ Y .Also,( , Y ) is said to be a subpermutation group of (, Y) if ≤ and Y is a union of orbits of acting on Y. The following is a useful well-known result on permutation groups whose proof is not difficult and is left as an exercise (see Exercise 1.10).
Theorem 1.1 Let (, Y) be a permutation group acting transitively on Y. Let y ∈ Y, let H = y be the stabiliser of y and let W be /H, the set of left cosets of H in . Then (, Y) is equivalent to (LH(), W). If (, Y) is not transitive, and O is the orbit containing y, then (LH(), W) is equivalent to the action of on Y restricted to O.
In the context of groups and graphs we shall need the very important idea of a group acting on pairs of elements of a set. Thus, let (, Y) be a permutation 1.2 Groups 5 group acting on the set Y.By(, Y × Y) we shall mean the action on ordered pairs of Y induced by as follows: If α ∈ and x, y ∈ Y, then
α((x, y)) = (α(x), α(y)). ( Y ) Similarly, by , 2 we shall mean the action on unordered pairs of distinct elements of Y induced by
α({x, y}) ={α(x), α(y)}.
These ideas will be developed further in a later chapter. In later chapters we shall also need the notions of k-transitivity and primitiv- ity of a permutation group. In order to study permutation groups in more detail one has to dig deeper into the concept of transitivity. Suppose, for example, that Y is the set {1, 2, 3, 4, 5} and is the group generated by the permutation α = (12345). Then clearly the permutation group (, Y) is transitive because for any i, j ∈ Y there is some power of α which maps i into j. But there is no power of α which, say, simultaneously maps 1 into 5 and 2 into 3. That is, not every ordered pair of distinct elements of Y can be mapped by a permutation in into any other given ordered pair of distinct elements. We therefore say that the permutation group (G, Y) is not 2-transitive. More generally, a permutation group (, Y) is said to be k-transitive if, given any two k-tuples (x1, x2, ..., xk) and (y1, y2, ..., yk) of distinct elements of Y, then there is an α ∈ such that
(α(x1), α(x2), ..., α(xk)) = (y1, y2, ..., yk).
Thus, a transitive permutation group is 1-transitive. Also, (, Y) is said to be k-homogeneous if, for any two k-subsets A, B of Y, there is an α ∈ such that α(A) = B, where α(A) ={α(a) : a ∈ A}. Finally, let (, Y) be transitive and suppose that R is an equivalence relation on Y, and let the equivalence classes of Y under R be Y1, Y2, ..., Yr. Then (, Y) is said to be compatible with R if, for any α ∈ and any equiv- alence class Yi,thesetα(Yi) is also an equivalence class. For example, if Y ={1, 2, 3, 4} and is the group generated by the permutation (1234), then (, Y) is compatible with the relation whose equivalence classes are {1, 3} and {2, 4}. Any permutation group is clearly compatible with the trivial equivalence relations on Y, namely, those in which either all of Y is an equivalence class or when each singleton set is an equivalence class. If these are the only equiva- lence relations with which (, Y) is compatible, then the permutation group is said to be primitive. Otherwise it is imprimitive. 6 Graphs and Groups: Preliminaries
If (, Y) is imprimitive and R is a nontrivial equivalence relation on Y with which the permutation group is compatible, then the equivalence classes of R are called imprimitivity blocks and their set Y/R is an imprimitivity block system for the permutation group (, Y). It is an easy exercise (see Exercise 1.14) to show that a 2-transitive permu- tation group is primitive. We shall also need some elementary ideas on the presentation of a group in terms of generators and relations. Let be a group and let X ⊆ .Aword in X is a product of a finite number of terms, each of which is an element of X or an inverse of an element of X. The set X is said to generate if every element in can be written as a word in X; in this case the elements of X are said to be generators of .Arelation in X is an equality between two words in X. By taking inverses, any relation can be written in the form w = 1, where w is some word in X. If X generates and every relation in can be deduced from one of the relations w1 = 1, w2 = 1, ... in X, then we write
= X|w1 = 1, w2 = 1, ... . This is called a presentation of in terms of generators and relations. The group is said to be finitely generated (respectively, finitely related)if|X| (respectively, the number of relations) is finite; it is called finitely presented, or we say that it has a finite presentation, if it is both finitely generated and finitely related. It is clear that every finite group has a finite presentation (although the con- verse is false). Simply take X = and, as relations, take all expressions of the form αiαj = αk for all αi, αj ∈ . In other words, the multiplication table of serves as the defining relations. It is well to point out that removing relations from a presentation of a group in general gives a larger group, the extreme case being that of the free group which has only generators and no relations. The simplest free group is the infinite cyclic group that has the presentation α with just one generator and no defining relation, whereas the cyclic group of order n has the presentation α|αn = 1 ; this group is denoted by Zn. The group with presentation α, β 1.3 Graphs and groups 7 is the infinite free group on two elements. The dihedral group of degree n is denoted by Dn. It has order 2n and also has a presentation with two generators: − − α, β|α2 = 1, βn = 1, α 1βα = β 1 .
Determining a group from a given presentation is not an easy problem. The reader who doubts this can try to show that the presentations
α, β : αβ 2 = β3α, βα2 = α3β and − − − α, β, γ : α3 = β3 = γ 3 = 1, αγ = γα 1, αβα 1 = βγβ 1 both give the trivial group. We shall of course make a very simple use of stan- dard group presentations where these difficulties do not arise. The book [159] is a standard reference for advanced work on group presentations. The reader is referred to [147, 222] for any terms and concepts on group theory that are used but not defined in these chapters and, in particular, to [49, 62] for more information on permutation groups.
1.3 Graphs and groups Let G, G be two graphs. A bijection α : V(G) → V(G ) is called an isomor- phism if
{u, v}∈E(G) ⇔{α(u), α(v)}∈E(G ). The graphs G, G are, in this case, said to be isomorphic, and this is denoted by G G . Similarly, if D, D are digraphs, then a bijection α : V(D) → V(D ) is called an isomorphism if
(u, v) ∈ A(D) ⇔ (α(u), α(v)) ∈ A(D ), and in this case the digraphs D, D arealsosaidtobeisomorphic, and again this is denoted by D D . If the two graphs, or digraphs, in this definition are the same, then α is said to be an automorphism of G or of D. The set of automorphisms of a graph or a digraph is a group under composition of functions, and it is denoted by Aut(G) or Aut(D). Note that an automorphism α of G is an element of SV(G), although it is its induced action on E(G) that determines whether α is an automorphism. This fact, although clear from the definition of automorphism, is worth emphasising when beginning to study automorphisms of graphs. 8 Graphs and Groups: Preliminaries
Figure 1.2. No automorphism permutes the edges as (12 23 34)
For example, for the graph in Figure 1.2, the permutation of edges given by (12 23 34) is not induced by any permutation of the vertex-set {1, 2, 3, 4}. The only automorphisms for this graph are the identity and the permutation (14)(23), which induces the permutation (12 34)(23) of the edges in the graph. The question of edge permutations not induced by vertex permutations will be considered in some more detail later in this chapter. The process of obtaining a permutation group from a digraph can be reversed in a very natural manner. Suppose that (, Y) is a group of permutations acting onasetY.LetA be a union of orbits of (, Y×Y). Clearly, the digraph D whose vertex-set is Y and whose arc-set is A has as a subgroup of its automorphism group. It might, however, happen that Aut(G) is larger than . Moreover, if the pairs in A are such that, for every (u, v) ∈ A, (v, u) is also in A, then replacing every opposite pair of arcs of D by a single edge gives a graph G such that ⊆ Aut(G). This and other ways of constructing graphs or digraphs admitting a given group of permutations will be studied in more detail in Chapter 4. Certain facts about automorphisms of graphs and digraphs are very easy to prove and are therefore left as exercises:
(i) Aut(G) = Aut(G); (ii) Aut(G) = SV(G) if and only if G or G is Kn, the complete graph on n vertices; (iii) Aut(Cn) = Dn. Also, let α be an automorphism of G and u, v vertices of G. Then,
(iv) deg(u) = deg(α(u)); (v) G − u G − α(u); (vi) d(u, v) = d(α(u), α(v)), where d(u, v) is the distance between u and v.
Also, if u is a vertex in a digraph D and α is an automorphism of D, then
(vii) degin(u) = degin(α(u)) and degout(u) = degout(α(u)). If u and v are vertices in a graph G and there is an automorphism α of G such that α(u) = v, then u and v aresaidtobesimilar.IfG − u G − v, then u and v aresaidtoberemoval-similar. Property (v) tells us that if two vertices are similar, then they are removal-similar. The converse of this is, however, 1.3 Graphs and groups 9 false, as can be seen from the graph shown in Figure 1.3. Here, the vertices u, v are removal-similar but not similar. Such vertices are called pseudosimi- lar. Similar, removal-similar and pseudosimilar edges are analogously defined: Two edges ab, cd of G are similar if there is an automorphism α of G such that α(a)α(b) = cd. We shall be studying pseudosimilarity in more detail in Chapter 5. Sometimes we ask questions of this type: how many graphs (possibly of some fixed order n) are there? The answer to this question depends heavily on how we consider two graphs to be different. In general, if the order of a graph G is n, we can think of its vertices as being labelled with the integers {1, 2, ..., n}. Two graphs G and H of order n so labelled are called identical or equal as labelled graphs (written G = H)if
ij ∈ E(G) ⇔ ij ∈ E(H).
(Compare this definition with that of isomorphic graphs.) Obviously, identical graphs are isomorphic, but the converse is not true. For example, the graphs in Figure 1.4 are isomorphic but not identical. Counting nonisomorphic graphs is, in general, much more difficult than counting nonidentical graphs. For example, there are four nonisomorphic graphs on three vertices but eight nonidentical ones. These are shown in Figures 1.5 and 1.6, respectively.
Figure 1.3. A pair of pseudosimilar vertices
Figure 1.4. Isomorphic but nonidentical graphs
Figure 1.5. The four nonisomorphic graphs of order 3 10 Graphs and Groups: Preliminaries
Figure 1.6. The eight nonidentical graphs of order 3
Counting nonisomorphic graphs involves consideration of group symme- tries. For more on this the reader is referred to [103].
1.4 Edge-automorphisms and line-graphs Although we shall be dealing mostly with Aut(G) and its realisation as the permutation group (Aut(G), V(G)), let us briefly look at other related groups associated with G. In this section we shall assume that G is a nontrivial graph, that is, its edge-set is nonempty. An edge-automorphism of a graph G is a bijection θ on E(G) such that two edges e, f are adjacent in G if and only if θ(e), θ(f ) are also adjacent in G.The set of all edge-automorphisms of G is a group under composition of functions, and it is denoted by Aut1(G). The concept of edge-automorphisms can perhaps be best understood within the context of line-graphs. The line-graph L(G) of a graph G is defined as the graph whose vertex-set is E(G) and in which two vertices are adjacent if and only if the corresponding edges are adjacent in G. An automorphism of L(G) is clearly an edge-automorphism of G and (Aut1(G), E(G)) is equivalent to (Aut(L(G), V(L(G))). In this section we shall give the exact relationship between Aut1(G) and Aut(G), that is, between the automorphism groups of G and L(G). As we described earlier, any automorphism α of G naturally induces a bijec- tion αˆ on E(G) defined by α(ˆ uv) = α(u)α(v). It is an important (and easy to 1.4 Edge-automorphisms and line-graphs 11 verify) property of αˆ that two edges e1, e2 are adjacent if and only if α(ˆ e1), α(ˆ e2) are adjacent, that is, if and only if αˆ is an edge-automorphism. For this reason αˆ is called an induced edge-automorphism of G. The set of all induced edge-automorphisms of G is denoted by Aut∗(G), and it is easy to check that this is a subgroup of Aut1(G) under composition of functions. Now, it seems natural to expect that Aut(G) and Aut∗(G) are iso- morphic. However, it can happen that two different automorphisms of G induce the same edge-automorphism. For example, let G = K2. Then |Aut(G)|=2 but |Aut∗(G)|=1. Also, suppose that G contains isolated vertices. Then any automorphism of G that permutes the isolated vertices and leaves all of the oth- ers fixed induces the trivial edge-automorphism. The following theorem says that these are basically the only situations when Aut(G) Aut∗(G).
Theorem 1.2 Let G be a nontrivial graph. Then Aut(G) Aut∗(G) if and only if G has at most one isolated vertex and K2 is not a component.
Proof Clearly, the mapping α →ˆα is a homomorphism from Aut(G) onto Aut∗(G) because αˆ .β(ˆ uv) = α.β(u)α.β(v) = αβ( uv). We must therefore show that the kernel of this mapping is trivial if and only if G has at most one isolated vertex and K2 is not a component. Suppose first that G has two isolated vertices u, v or K2 as a component with vertices u, v. Then the permutation α that transposes u and v and fixes all of the other vertices is a nontrivial automorphism of G,butαˆ is the identity. Therefore the kernel is not trivial. Conversely, suppose that G does not contain K2 as a component nor its com- plement. If Aut(G) is trivial, then so is Aut∗(G). Therefore, let α be a nontrivial element of Aut(G), and let α(u) = v = u. Then deg(u) = deg(v) = 0 (other- wise u, v would be a pair of isolated vertices). We consider two cases. Case 1: u, v adjacent. Let e be the edge uv. Then deg(u) = deg(v)>1 (otherwise the two vertices u, v would form a component K2). Therefore, there exists an edge f = e incident to u (but not to v, since the graph is simple). But α(ˆ f ) must be incident to v (since α(u) = v), that is, α(ˆ f ) = f , and hence αˆ is not trivial. Case 2: u, v not adjacent. Let e be an edge incident to u.Again,e is not incident to v but α(ˆ e) is. Therefore αˆ is again nontrivial.
The next natural question to ask is whether there can be edge-automorphisms of G that are not induced by automorphisms, that is, whether Aut∗(G) can be a strict subgroup of Aut1(G). This situation can very well happen, although, as we shall see, such cases are quite rare. 12 Graphs and Groups: Preliminaries
Figure 1.7. Graphs with edge-isomorphisms not induced by isomorphisms
Before proceeding let us first extend the idea of edge-automorphisms on the edge-set of a graph to that of edge-isomorphisms between edge-sets of different graphs. Let G, G be two nontrivial graphs. A bijection θ : E(G) → E(G ) is an edge-isomorphism if
e, f adjacent in G ⇔ θ(e), θ(f ) adjacent in G .
Two graphs are said to be edge-isomorphic if there is an edge-isomorphism between their edge-sets. The graphs W1, W2 in Figure 1.7 are edge-isomorphic, although they are not isomorphic. That is, there is an edge-isomorphism between their edge- sets that cannot be induced by an isomorphism between their vertex-sets. This means that their line-graphs are isomorphic even though the two graphs are not themselves isomorphic. Also, each of the graphs W3, W4, W5 in the same figure has edge- automorphisms that are not induced by automorphisms. That is, the group ∗ Aut (Wi) is a strict subgroup of Aut1(Wi). In other words, Aut(L(Wi)) is larger than Aut(Wi). The following theorem of Whitney [258] says that these are essentially the only cases when edge-isomorphisms that are not induced by isomorphisms can arise. We give the statement of the theorem without proof, which, although not deep or difficult, would lengthen this introductory chapter without adding significant new insights.
Theorem 1.3 (Whitney) Let G, G be connected graphs different from the five graphs in Figure 1.7. Let θ : E(G) → E(G ) be an edge-isomorphism. Then θ is induced by an isomorphism from G to G . 1.5 A word on issues of computational complexity 13
Whitney’s Theorem and Theorem 1.2 together therefore give the following corollary.
∗ Corollary 1.4 Let G be a nontrivial graph. Then Aut1(G) = Aut (G) if and only if both of these conditions hold:
(i) not both W1, W2 are components of G; (ii) none of Wi, i = 3, 4, 5 is components of G.
Moreover, Aut1(G) Aut(G) (that is, Aut(L(G)) Aut(G)) if and only if (i) and (ii) hold and G has at most one isolated vertex and K2 is not a component of G.
1.5 A word on issues of computational complexity Although in this book we shall not concern ourselves with issues of computa- tional complexity, it is perhaps worthwhile to say a few words in this regard here in order to put matters into a better perspective. A student reading the def- initions of isomorphic graphs and automorphisms might think that it is an easy matter to determine in general whether two given graphs are isomorphic or to compute the automorphism group of a graph. In fact, this is far from being the case, and these problems are very hard to crack in practice, at least as far as present knowledge goes. In general, one considers that an efficient algorithm exists for finding a solu- tion to a problem (for example, finding a nontrivial automorphism of a given graph) if there is a general algorithm such that the number of operations that it takes to solve the problem is a polynomial function of the size of the input (say, the number of vertices in the graph); one says that the algorithm solves the problem in polynomial time. Of course, several terms in the previous sentence need exact definitions, but we shall here take an intuitive approach and refer the reader to [36]or[82]for the exact details on computational complexity. Those problems for which an efficient (polynomial-time) algorithm exists form the class denoted by P (which stands for ‘polynomial’). However, there are several problems for which it is not known whether an efficient algorithm does exist. In order to tackle this question of computational intractibility, two important ideas have been developed. Firstly, the class NP (which stands for ‘nondeterministic polynomial’) is defined. Roughly (again we refer the reader to the textbooks cited earlier for the exact details) this class contains all of those problems for which, given a candidate solution, one can verify in polynomial time that it is in fact a correct 14 Graphs and Groups: Preliminaries solution. For example, the problem of determining whether a graph has a non- trivial automorphism is in NP, since, given such a permutation of the vertices, it is easy to determine in polynomial time that it is an automorphism. Now the main question in computational complexity is whether P = NP (clearly P ⊆ NP), and to tackle this question another important idea is intro- duced. Given two problems A and B, one says that A is (polynomially) reducible to B if, given an algorithm for solving B, it can be transformed in polynomial time into an algorithm for solving A. Reducibility therefore introduces a hier- archy between problems for, if A is reducible to B, then, in a sense, A cannot be more difficult to solve (computationally) than B. In particular, if there is an efficient algorithm for solving B, then there is also an efficient algorithm for solving A. Now, the question of reducibility took on special significance by the discov- ery that in the class NP there are problems, called NP-complete, to which any other problem in NP is reducible. In other words, if an efficient algorithm can be found for any NP-complete problem, then all problems in NP would have an efficient algorithm to solve them, and P would be equal to NP. Now, it is not known whether the problem of determining if two graphs are isomorphic, which lies clearly in NP,isNP-complete. In fact, if it turns out that P is not equal to NP, then there is evidence to suggest that the problem of graph isomorphism might lie strictly between the classes P and NP. What all this means in practice is that, as far as present knowledge goes, no general algorithm can determine in a guaranteed reasonable time whether two graphs are isomorphic, or whether a given graph has a nontrivial automorphism (these two problems are closely related [126]). It is known that for special types of graphs (for example, trees, planar graphs and graphs with bounded degree) an efficient algorithm does exist. Computer packages can also help one to solve these problems, certainly more efficiently than an attempt ‘by hand’ for large graphs, although their time performance is not guaranteed (by what we said earlier). For example, the soft- ware package MathematicaTM has a combinatorics extension that, amongst other things, finds graph automorphisms and isomorphisms. A more specialised package, and one that is freely available from www.combinatorialmath.org.ca/g&g/index.html is Groups & Graphs [131], developed by Bill Kocay. This package contains several combinatorial routines related to graphs, digraphs, combinatorial designs and their automorphism groups and also embeddings of graphs on some sur- faces and a graph isomorphism algorithm. It is easy to use, and it has a pleasant graphical user interface. It is also very useful simply for drawing diagrams of 1.6 Exercises 15 graphs. Although originally written for MacintoshTM computers, a version for the unix-based Haiku operating system is in preparation, and this version will contain several new features. An important computer algebra package, which is also freely available, is the system GAP [243]. This package performs very sophisticated routines in discrete abstract algebra, in particular routines on permutation groups. It incor- porates a number of extensions, one of which, GRAPE [235], deals specifically with graphs, including their automorphisms and isomorphisms. The computer package Sage [227] is an open-source competitor to systems like MapleTM, MathematicaTM and MatlabTM. It incorporates several open- source mathematical software like GAP and R, and it can be run via Sage- MathCloud without the need of installing the system on one’s computer. It has an excellent library of functions for doing graph theory. In this book we shall present some constructions using GAP and Sage. Finally, it should be mentioned that it is generally accepted that the best package to tackle graph isomorphisms is nauty [181], developed by Brendan McKay. In fact, the system GRAPE invokes nauty when computing automor- phisms or isomorphisms.
1.6 Exercises 1.1 Draw all twenty nonidentical graphs with vertex-set {1, 2, 3, 4} that have three edges. How many of them are nonisomorphic? In general, how many nonidentical graphs on n vertices and m edges are there? How many are there on n vertices? 1.2 Let G be the graph in Figure 1.8. How many nonidentical labellings does G have using the labels {1, 2, ...,6} on its vertices? In general, how many nonidentical labellings does a graph G on n vertices have using the labels {1, 2, ..., n} on its vertices? 1.3 Show that if G is self-complementary (that is, G G), then n ≡ 0 mod 4 or n ≡ 1 mod 4. Determine all self-complementary graphs on five vertices. 1.4 A well-known result due to Cayley says that the number of nonidentical trees on − n vertices is nn 2. Verify this for n = 4. Look up one of the several proofs of this result.
Figure 1.8. How many distinct labellings does this graph have? 16 Graphs and Groups: Preliminaries
The points 1, 2, ..., n are drawn in a plane. A random tree is drawn joining these points, with all possible spanning trees being equally likely. Let pn be the probability that 1 is an endvertex of the tree. Show that limn→∞ pn = 1/e. 1.5 Find a graph with a pair of pseudosimilar edges. 1.6 Let Y ={1, 2, 3, 4} and let be the group acting on Y generated by the permu- tation (1234). Construct a digraph D whose vertex-set is Y and whose arc-set is the orbit of the arc (1, 2) under the action of the permutation group (, Y × Y). Is the whole of Aut(D)? Can a graph be obtained by taking the orbit of some other arc or a union of orbits? Will always be the whole of Aut(D)? 1.7 Let P be a rectangular plate and Q a plate in the form of a rhombus. Show that the groups of symmetry of P and Q are isomorphic as abstract groups but not equivalent considered as permutation groups of the four vertices of P and Q. Show that the same situation arises with the following two graphs: the cycle on four vertices with an extra multiple edge and the complete graph K4 with an edge deleted. 1.8 Show that the dihedral group Dn can be presented as α, β|α2 = β2 = (αβ)n = 1 .
1.9 Let 1 be the abelian group defined by the presentation α, β, γ |α3 = β3 = γ 3 = 1, [α, β]=[α, γ ]=[β, γ ]=1 , −1 −1 where [α, β]=α β αβ is the commutator of α and β, and let 2 be the group defined by the presentation α, β, γ |α3 = β3 = γ 3 = 1, [β, α]=γ , [α, γ ]=[β, γ ]=1 .
Show that although 1 and 2 are not isomorphic as abstract groups, and there- fore the two permutation groups (L(1), 1) and (L(2), 2) are not equivalent, still the cycle structures of the permutations of L(1) acting on 1 are the same as those of the permutations of L(2) acting on 2. Give an example of two permutation groups whose permutations have the same cycle structures and which are isomorphic as abstract groups but are still not equivalent as permutation groups. 1.10 Prove Theorem 1.1. 1.11 This exercise is intended to illustrate Theorem 1.1. Consider the action of the alternating group A4 on the set X ={1, 2, 3, 4}.LetH be the stabiliser of the element 1 under this action. Confirm that the left action of A4 on the left cosets of H is equivalent to the action of A4 on X. 1.12 Let be a group of permutations acting on a set Y and let y, x be two elements of Y that are in the same orbit under this action. Let α, β ∈ be two permutations such that α(y) = β(y) = x. Prove that αy = βy.
Prove also that if x is in the same orbit as y and γ is a permutation such that
γ(y) = x and γy = αy,thenx = x. 1.13 Show that if the abelian permutation group acts transitively on Y, then its action on Y is regular. 1.14 (a) Show that a 2-transitive permutation group is primitive. (b) Show that if Aut(G), for a graph G, is 2-transitive, then either G or its com- plement is a complete graph. 1.7 Notes and guide to references 17
(c) Suppose that the transitive permutation group (, Y), with |Y| finite, is imprim- itive. Show that the blocks of an imprimitivity block system have equal size. 1.15 Let G be a vertex-transitive graph whose automorphism group acts impri- mitively on V(G). Show that the subgraphs of G induced by the blocks of an imprimitivity block system are all isomorphic. Suppose that each such subgraph is replaced by its complement, leaving the
other edges intact. Let G be the resulting graph. Show that Aut(G ) = Aut(G). 1.16 The Petersen graph can be defined as follows. Let N ={1, 2, 3, 4, 5}, and let the vertices of the graph be all subsets of N of size 2 in which two vertices are adja- cent if the corresponding subsets are disjoint. Use this definition and GAP (with GRAPE) to construct the Petersen graph and to verify some of its properties.
1.7 Notes and guide to references One of the standard texts on graph theory has, for many years, been [97]. More recent books that give an excellent coverage of the subject are [28, 61, 257, 259]. The last reference is a short introduction that is quite sufficient back- ground for this book. Biggs’ book [24] is the standard text on algebraic graph theory, but the more recent [90] is also an excellent and up-to-date textbook on the subject. The book [94] contains a number of recent and specialised survey papers on various aspects of algebraic graph theory, particularly those dealing with graph symmetries. A proof of Whitney’s Theorem can be found in [22]. We shall need only the most elementary notions of group theory. The text [147] gives ample coverage for our purposes, while [222] provides a more complete treatment. Two excellent books devoted entirely to permutation groups are [49, 62]. Most of the results and definitions on permutation groups that we have given here and others that we shall need can be found in the first few chapters of these two books. For a full discussion of the terms on computational complexity that were introduced earlier rather intuitively, the reader is referred to the standard text- book [82] or the more recent [36]. The book [126] and the references it cites are suggested for those who are interested in the computational complexity of the graph isomorphism problem. Those who are particularly interested in some of the powerful algebraic techniques used to tackle this problem should look at the papers [106, 155]. For practical computations on a computer with per- mutation groups and graph automorphisms and isomorphisms in particular, the systems [131, 181, 235, 243] are recommended. 2 Various Types of Graph Symmetry
We shall see in this chapter that most graphs are asymmetric, that is, their automorphism group is trivial; in other words, it consists only of the identity permutation. The least number of vertices that an asymmetric graph can have is six, and the graph shown in Figure 2.1 is the smallest such graph in the sense that any other asymmetric graph on six vertices has more edges. As is to be expected, however, the most interesting relationships between groups and graphs arise when the graphs have a very high degree of symmetry, that is, a large automorphism group. One way to make more precise the idea of a large automorphism group is to require that it at least be transitive on the vertex-set or the edge-set of the graph. The weakest forms of symmetry to ask of a graph involve vertex-transitivity and edge-transitivity, which we define in the next section.
2.1 Transitivity Recall the definition of similar vertices from the previous chapter. We say that a graph G is vertex-transitive if any two vertices of G are similar, that is, if, for any u, v ∈ V(G), there is an automorphism α of G such that α(u) = v. In other words, G is vertex-transitive if all of the vertices of G are in the same orbit of the permutation group (Aut(G), V(G)).
Figure 2.1. The smallest asymmetric graph
18 2.1 Transitivity 19
Figure 2.2. A vertex-transitive graph that is not edge-transitive
One can define edge-transitivity analogously. A graph G is edge-transitive if, given any two edges {a, b} and {c, d}, there exists an automorphism α such that α{a, b}={c, d}, that is, {α(a), α(b)}={c, d}. In other words, G is edge- transitive if any two of its edges are similar under the action of the permutation group (Aut∗(G), E(G)), that is, if the edges of G are all in one orbit under this action. Note that very often the word ‘transitive’ is used to refer to a graph, and in this case it is taken to mean ‘vertex-transitive’. Vertex-transitivity does not imply edge-transitivity, nor does the converse implication hold. Figure 2.2 shows a graph that is vertex-transitive but not edge-transitive. The complete bipartite graph Kp,q with p = q is a simple example of a graph that is edge-transitive but not vertex-transitive. The following well-known result does give a description of edge-transitive graphs that are not vertex- transitive.
Theorem 2.1 Let G be a graph without isolated vertices and let H be a sub- group of Aut(G). Suppose that the action induced by H is transitive on the edges of G but not on its vertices. Then G is bipartite and the action of H on V(G) has two orbits that form the bipartition of V(G).
Proof Let {u, v} be an edge of G.LetV1, V2 be the orbits under the action of H containing u and v, respectively. (We are not excluding, for the moment, the possibility that V1 = V2.) Let x be any other vertex of G. Since G does not have isolated vertices, there exists a vertex y adjacent to x; that is, {x, y} is an edge of G.ButG is edge-transitive under the action of H; therefore the two edges are similar under this action. Hence x is similar to at least one of u or v, that is, x is in V1 or V2. Therefore V1 ∪ V2 = V(G). Now, V1, V2 must be disjoint; otherwise (since orbits form a partition) they are equal, and this would mean that the vertices of G are all in one orbit, giving that G is vertex-transitive under the action of H. Hence we now have that the action of H on V(G) has exactly two orbits, V1, V2. 20 Various Types of Graph Symmetry
Now let a, b be in the same orbit, say V1. It then follows that a, b are not adjacent. For suppose otherwise. Then the edge {u, v} is similar to the edge {a, b} under the action of H; therefore the vertex v is similar to one of a or b under this action, giving that v (which is in the orbit V2) is also in the orbit V1, which contains a and b. But this is impossible since V1 ∩ V2 =∅. Hence, as required, we have that no two vertices from the same orbit can be adjacent.
Corollary 2.2 If a graph G without isolated vertices is edge-transitive but not vertex-transitive, then it is bipartite and the action of Aut(G) on V(G) has two orbits that form the bipartition of V(G).
Proof Take H = Aut(G) in the previous theorem.
2.1.1 Semisymmetric graphs The typical example of edge-transitive but not vertex-transitive graphs given earlier is the complete bipartite graph with a different number of vertices in the bipartition. These graphs are trivially not vertex-transitive because their vertices have different degrees. Although regular graphs do exist that are edge- transitive but not vertex-transitive, it is quite difficult to construct them. Such graphs are now called semisymmetric graphs and they were first studied by Folkman [77], who constructed the smallest possible semisymmetric graph having twenty vertices.1 One construction of the Folkman Graph is described in Exercise 2.5. Here we shall describe another well-known semisymmetric graph, the Gray Graph.2 For a long time nobody could find a smaller cubic semisymmetric graph, and eventually it was formally proved in [160] that it is the smallest cubic semisymmetric graph. We shall see a more systematic way of describing it in subsequent chapters. Here we follow Bouwer’s construction in [33]. Consider a cycle on 54 vertices numbered consecutively from 0 to 53. To form the Gray Graph G add the following edges to this cycle: {1, 42}, {2, 15}, {3, 28}, {4, 33}, {5, 44}, {6, 53}, {7, 48}, {8, 21}, {9, 32}, {10, 45}, {11, 24}, {12, 41}, {13, 20}, {14, 31}, {16, 35}, {17, 40}, {18, 49}, {19, 0}, {22, 51}, {23, 30}, {25, 38}, {26, 43}, {27, 52}, {29, 36}, {34, 47},
1 More information about this graph, called the Folkman Graph, can be found on the MathWorld page http://mathworld. wolfram.com/FolkmanGraph.html 2 The Gray Graph is also featured on MathWorld at http://mathworld.wolfram.com/Gray Graph.html 2.1 Transitivity 21
{37, 50}, {39, 46}.
It is tedious but not difficult to check that the permutations
α = (2043)(35343)(4644)(74533)(81032)(11 31 21) = (12 14 20)(15 19 41)(16 18 40)(22 24 30)(25 29 51) = (26 28 52)(34 48 46)(35 49 39)(36 50 38) and
β = (171137155392535)(2 6 10 38 16 0 8 24 36) = (3545391719242329)(44446401820223028) = (12 50 14 52 32 26 34 42 48)(13 51 31 27 33 43 47 41 49) are automorphisms of G. Note that the automorphism α fixes the vertex 1 and permutes cyclically its neighbours 2, 42 and 0. Thus, in order to show that the graph is edge-transitive it is sufficient to show that any odd-numbered vertex can be mapped into 1 by an automorphism of G. This can be done by appropriate products of α and β. For example, α4β maps vertex 53 to vertex 1. However, the graph is not vertex-transitive because from an odd-numbered vertex it is possible to have three different paths of length 4 joining the vertex to some other common vertex (for example, vertex 1 to vertex 5), but this is not possible from an even-numbered vertex. Another way to show that the Gray Graph is not vertex-transitive is to con- sider the distance sequences of its vertices [166]. The distance sequence of a vertex v is the vector (a0, a1, ..., ar) where ai is the number of vertices at dis- tance i from v. In the case of the Gray Graph, the distance sequences of the vertices in the two colour classes are (1, 3, 6, 12, 12, 12, 8) and (1, 3, 6, 12, 16, 12, 4), respectively, therefore these vertices cannot be in the same orbit under the automorphism group of the graph. Although the Sage package has the Gray Graph already implemented, it is easy and instructive to show how to construct it following the specification given earlier. First one creates the vertex-set which will be the list of numbers from 1 to 54. Sage, like many computer languages such as Python, starts its lists from 0. Therefore a list of length n produced by the command range(55) would contain the numbers from 0 to 54. To start the list from 1 we have to define the list of vertices as
vertices := range(1,55); 22 Various Types of Graph Symmetry
The graph is then constructed first using the command DiGraph so that we do not need to repeat every pair of adjacent vertices twice. This com- mands basically takes two parameters. The first parameter is the vertex-set of the graph to be constructed, and the second parameter is a boolean function (defined with the lambda construct) of two variables which compares all pos- sible pairs of the vertex-set and an edge is drawn between any pair of vertices for which the function returns True. dgray := DiGraph([vertices, lambda i, j: (i == mod[j + 1, 54]) or (i == 53 and j == 54) or (i == 1 and j == 42) or (i == 2 and j == 15) or (i == 3 and j == 28) or (i == 4 and j == 33) or (i == 5 and j == 44) or (i == 6 and j == 53) or (i == 7 and j == 48) or (i == 8 and j == 21) or (i == 9 and j == 32) or (i == 10 and j == 45) or (i == 11 and j == 24) or (i == 12 and j == 41) or (i == 13 and j == 20) or (i == 14 and j == 31) or (i == 16 and j == 35) or (i == 17 and j == 40) or (i == 18 and j == 49) or (i == 19 and j == 54 or (i == 22 and j == 51) or (i == 23 and j == 30) or (i == 25 and j == 38) or (i == 26 and j == 43) or (i == 27 and j == 52) or (i == 29 and j == 36) or (i == 34 and j == 47) or (i == 37 and j == 50) or (i == 39 and j == 46) ] ) This digraph is then changed into an undirected graph with the following command which changes every arc into an edge. 2.1 Transitivity 23
gray = dgray.to_undirected()
It is then easy to check that the aforementioned properties of the Gray Graph hold. For example, in order to check whether it is vertex-transitive we use the command
gray.is_vertex_transitive() which returns False. The command
gray.is_edge_transitive() returns True, as expected, while the command
gray.is_regular() also returns True, confirming that the graph is semisymmetric. In fact, we could have reached the same conclusion with the command
gray.is_semi_symmetric() which also returns True. Finally, one can check whether the graph constructed earlier is isomorphic to Sage’s inbuilt ‘GrayGraph’ using the command graphs.GrayGraph.is_isomorphic(gray) which, again, returns True. We shall have more to say about the Gray Graph in a later chapter when we shall describe it in a more algebraic fashion. Exercises 2.2 and 2.5 show that any semisymmetric graph must have even order and its degree must be less than |V(G)|/2.
1 2.1.2 Arc-transitive and 2 -arc-transitive graphs A stronger form of transitivity than either vertex- or edge-transitivity based on the edge-set of G can also be defined. If G has the property that, for any two edges {a, b}, {c, d}, there is an automorphism α such that α(a) = c and α(b) = d and also an automorphism β such that β(a) = d and β(b) = c, then G is said to be arc-transitive. We shall now derive a result of Tutte that gives a restriction on the degree of the vertices of a graph that is vertex-transitive and edge-transitive but not arc-transitive. One can think←→ of arc-transitivity as follows. Given any graph G, construct the directed graph G obtained from G by replacing each edge {a, b} by the pair of 24 Various Types of Graph Symmetry ←→ arcs (a, b) and (b, a). Then clearly Aut(G) = Aut( G ) and any automorphism α of G induces the natural action on arcs given by
(a, b) → (α(a), α(b)). ←→ Then G is arc-transitive←→ precisely if, given any two arcs in G , there is an automorphism of G mapping one arc into the other. This is a stronger form of transitivity than both vertex- and edge-transitivity because now, given any two edges on each of which an orientation is imposed, there is an automorphism mapping one edge into the other and preserving the given orientations. In fact, an arc-transitive graph is both vertex-transitive and edge-transitive.
Lemma 2.3 Let H be a subgroup of Aut(G) such that, under the action of H,G ←is→ vertex-transitive and edge-transitive←→ but not arc-transitive. Let←→ t be an arc of G and let D be the subdigraph of G whose vertex-set is V( G ) and whose arc-set is the orbit of t under the action of H. Then (i) for every edge {a, b} of G, D contains exactly one of the arcs (a, b) or (b, a); (ii) H ≤ Aut(D); (iii) D is vertex-transitive.
Proof (i) Let t = (s1, s2). Because G is edge-transitive under the action of H, there is an α ∈ H such that α{s1, s2}={a, b}. Therefore certainly one of (a, b) or (b, a) is an arc of D. Suppose that both are arcs of D. Then there is some β ∈ H such that β((a, b)) = (b, a). But given any edge {c, d} of G there is, by edge-transitivity, a γ ∈ H such that γ((a, b)) equals (c, d) or (d, c). Suppose, without loss of generality, that γ((a, b)) = (c, d). But then γβ((a, b)) = (d, c). Therefore, for any edge {c, d} of G, both arcs (c, d) and (d, c) areinthesame orbit, that is, the action of H on G is arc-transitive, a contradiction. (ii) This follows because the arc-set of D is a full orbit of the permutation group (H, V(G) × V(G)). (iii) This follows because (H, V(G)) is transitive, V(D) = V(G) and H ≤ Aut(D). If we let H = Aut(G) in this lemma, then, in view of (i), if G is a vertex- transitive and←→ edge-transitive graph that is not arc-transitive, it follows that the arc-set of G is naturally partitioned into two orbits of equal size under the action of Aut(G), and none of the two orbits contains both an arc (a, b) and its inverse (b, a). In view of this, a graph that is vertex-transitive, edge-transitive 1 but not arc-transitive is said to be 2 -arc-transitive. 2.2 Asymmetric graphs 25
Figure 2.3. Relationship between different types of transitivity
The relationship between these forms of transitivity is shown in Figure 2.3, where a line leading down from one property to another means that the first implies the second. 1 Although 2 -arc-transitive graphs are not easy to find, they do exist (an exam- ple will be given in Chapter 3). The following well-known theorem of Tutte tells us that such a graph must have even degree.
Theorem 2.4 (Tutte) Let H be a subgroup of Aut(G) such that, under the action of H, G is vertex-transitive and edge-transitive but not arc-transitive. 1 Then the degree of G is even. In particular, a 2 -arc-transitive graph has even degree.
Proof Let D be as in the previous lemma. By the third part of this lemma, all vertices of D have the same out-degree, say k.Now,k·|V(D)|=|A(D)| and, by the first part of the lemma, |A(D)|=|E(G)|. But if the common degree of the vertices of G is d, then, by the Handshaking Lemma, |E(G)|=d ·|V(G)|/2 = d ·|V(D)|/2. Therefore d = 2k, that is, d is even.
2.2 Asymmetric graphs Although we shall be mostly interested in graphs with nontrivial automorphism groups, let us briefly consider asymmetric graphs. Let P be a graph theoretic property such as ‘planar’ or ‘vertex-transitive’. Let rn denote the proportion of 26 Various Types of Graph Symmetry labelled graphs on n vertices that have property P. If limn→∞ rn = 1, then we say that almost every (a.e.) graph has property P. We have already said that almost every graph is asymmetric. We shall soon prove a stronger result that will be used in a later chapter when we consider the Reconstruction Problem. The following probability space is often set up when studying random graphs. Let G(n, p) be the set of all labelled graphs on the set of vertices {1, 2, ..., n} where, for each pair i, j, P(ij is an edge) = p and P(ij is not an edge) = 1 − p independently. Therefore a graph with m edges in G(n, p) has probability pm (n)−m q 2 , where q = 1 − p. We shall need only this space when the probability (n) = 1 G( 1 ) ( 1 ) 2 p 2 . In this case, each graph G in n, 2 has probability 2 , which is, of course, equal to the probability of choosing G randomly from amongst (n) all 2 2 labelled graphs on n vertices when all are equally likely to be chosen. Therefore, to show that a.e. graph has a particular property P one has to show ∈ G( 1 ) P that the probability that G n, 2 has property tends to 1 as n tends to infinity. Now, let k be fixed. We say that a graph G has property Ak if all induced subgraphs of G on n − k vertices are mutually nonisomorphic. In other words, G has property Ak means that, if X, Y are two distinct k-subsets of V(G), then G − X G − Y. It is easy to show (Exercise 2.5) that if G has property Ak+1, then it also has property Ak and that if it has property A1, then it is asymmetric. We shall show that, for any fixed k, a.e. graph has property Ak.
Lemma 2.5 Let W ⊆ V, |W|=t, |V|=n, and let ρ : W → Vbeaninjective function that is not the identity. Let g = g(ρ) be the number of elements w ∈ W such that ρ(w) = w. Then there is a set Iρ of pairs of (distinct) elements of W, containing at least 2g(t − 2)/6 pairs, such that Iρ ∩ ρ(Iρ) =∅.
Proof Consider those pairs v, w ∈ W such that at least one is moved. (All pairs ( − ) + g are taken to contain distinct elements.) There are g t g 2 such pairs. For all but at most g/2 of these pairs, {v, w} ={ρ(v), ρ(w)} (the exceptions are when ρ(v) = w and ρ(w) = v). Let Eρ be the set of all such pairs. Then g |Eρ|≥g(t − g) + − g/2 = g(t − g/2 − 1) ≥ g(t/2 − 1). 2 2.2 Asymmetric graphs 27
Define a graph Hρ with vertex-set the pairs in Eρ and such that each pair {v, w} is adjacent to the pair {ρ(v), ρ(w)}.InHρ, all degrees are at most 2. Degrees equal to 1 could arise because {ρ(v), ρ(w)} could contain an element not in W, and so the pair would not be in Eρ. Degrees equal to 2 could arise because {v, w} could be adjacent to both {ρ(v), ρ(w)} and {ρ−1(v), ρ−1(w)}. Therefore the components of Hρ are isolated vertices, paths or cycles. Let Iρ be a set of independent (that is, mutually not adjacent) vertices in Hρ. There- fore, for any pair {v, w}∈Iρ, {ρ(v), ρ(w)} is not in Iρ. Now, all isolated vertices in Hρ are independent, at least half of the vertices on a path are independent and at least one third of the vertices on a cycle are independent, the extreme case here being a triangle. Therefore
|Iρ|≥|Eρ|/3 ≥ 2g(t − 2)/6, as required.
∈ G( 1 ) ⊂ = ( ) | |= ρ → Corollary 2.6 Let G n, 2 ,W V V G and W t. Let : W V be an injective function that is not the identity. Let g = g(ρ) be the number of elements w ∈ W such that ρ(w) = w. Let Sρ be the event ‘ρ gives an isomorphism from G[W] to G[ρ(W)]’.
Then ( − )/ 1 2g t 2 6 P(Sρ) ≤ . 2
Proof Let Iρ be the set constructed in the previous lemma. Now, for a given pair {v, w}∈Iρ, the event
‘{v, w} and {ρ(v), ρ(w)} are both edges or nonedges’ has probability 1/2. These events, as they range over all pairs {v, w}∈Iρ,are mutually independent because they involve distinct pairs. But Sρ requires all these events simultaneously. Therefore, by independence, | | ( − )/ 1 Iρ 1 2g t 2 6 P(Sρ) ≤ ≤ , 2 2 as required. The result of this corollary is the crux of the proof of the following theo- rem: There are too many independent correct ‘hits’ required for ρ to be an isomorphism, and the probability therefore becomes small as n increases. 28 Various Types of Graph Symmetry
Theorem 2.7 (Korshunov; Muller;¨ Bollobas)´ Let k be a fixed nonnegative ∈ G( 1 ) integer and let G n, 2 . Let pn denote the probability that
∃W ⊆ V(G) = V ={1, 2, ..., n}, with |W|=n − k and such that
∃ρ : W → V, ρ = id, ρ is an isomorphism from G[W] to G[ρ(W)].
Then, limn→∞ pn = 0.
Hence, a.e. graph has property Ak.