Finite s-geodesic Transitive Graphs. Wei Jin

Transcription

1 Finite s-geodesic Transitive Graphs Wei Jin This thesis is presented for the degree of Doctor of Philosophy of The University of Western Australia School of Mathematics and Statistics May 8, 2013

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3 Abstract A geodesic from a vertex u to a vertex v in a graph is one of the shortest paths from u to v, and this geodesic is called an s-geodesic if the distance between u and v is s. For a graph Γ and for an integer s less than or equal to the diameter of Γ, assume that for each i s, all i-geodesics of Γ are equivalent under the group of graph automorphisms. The purpose of this thesis is to study such graphs, called s-geodesic transitive graphs. In the first part of the thesis, we show that the subgraph [Γ(u)] induced on the set of vertices of Γ adjacent to a vertex u is either (i) a connected graph of diameter 2, or (ii) a union mk r of m 2 copies of a complete graph K r with r 1. This suggests a way forward for studying s-geodesic transitive graphs according to the structure of such graphs [Γ(u)]. We study further the family F(m, r) of connected graphs Γ such that [Γ(u)] = mk r for each vertex u, and for fixed m 2, r 1. We show that each Γ F(m, r) is the point graph of a partial linear space S of order (m, r + 1) which contains no triangles. Conversely, each S with these properties has point graph in F(m, r), and a natural duality on partial linear spaces induces a bijection F(m, r) F(r + 1, m 1). In the second part of the thesis, we compare 2-geodesic transitivity of graphs with another two transitivity properties, namely, 2-distance transitivity and 2-arc transitivity. It is easy to verify that if a non-complete graph is 2-arc transitive, then it is 2-geodesic transitive, which in turn implies that it is 2-distance transitive. We classify 2-geodesic transitive but not 2-arc transitive graphs of valency 4 and prime valency, and we also prove that, (except for a few cases) the Paley graphs and the Peisert graphs are 2-distance transitive but not 2-geodesic transitive. In the third part of the thesis, we prove reduction theorems for the family of s-geodesic transitive graphs with [Γ(u)] connected, and the family whose [Γ(u)] is disconnected. In each case, we identify a subfamily of basic s-geodesic transitive graphs such that each s-geodesic transitive graph has at least one basic s-geodesic 3

4 4 transitive graph as a normal quotient. We study such basic graphs where a group G of graph automorphisms is quasiprimitive on the vertex set. Many of these basic graphs are Cayley graphs. This leads us to study (G, 2)-geodesic transitive Cayley graphs Cay(T, S) with G contained in the holomorph of T. This study can be further reduced to the following three problems: investigating the case that T is a minimal normal subgroup of G, studying the 2-geodesic transitive covers of these graphs, and investigating the 2-geodesic transitive covers of complete graphs.

5 Contents Abstract 3 Acknowledgements 7 Publications arising from this thesis 9 List of key symbols 11 1 Introduction The aims and main results of the thesis Literature review Layout of thesis Notation, definitions and preliminary results Finite group theory Permutation groups Primitive groups and quasiprimitive groups Graph theory Basic concepts Symmetry properties of graphs Partial linear spaces Locally 2-geodesic transitive graphs Overview and main results Local structures Disconnected neighbourhoods Clique graphs Partial linear spaces Connected neighbourhoods

6 6 CONTENTS 3.5 A reduction Disconnected neighbourhoods Connected neighbourhoods Comparison of distance, geodesic and arc transitivity for graphs Overview and main results Geodesic transitive graphs Johnson graphs Hamming graphs Odd graphs s-arc transitivity Paley graphs and Peisert graphs Geodesic transitive graphs of prime valency The main results Arc-transitive graphs of odd prime order Properties of graphs in C(p) Prime valency 2-geodesic transitive graphs Line graphs and s-geodesic transitivity Overview and main results Line graphs The map L s Proof of Theorem Geodesic transitive graphs of valency at most Geodesic transitive Cayley graphs Overview and main results Proof of Theorem A reduction theorem Complete multipartite graphs Bibliography 115

7 Acknowledgements I would first like to express my sincere gratitude to my supervisor, Winthrop Professor Cheryl E. Praeger, for her invaluable supervision, patient guidance and continuous encouragement throughout the preparation of the thesis. It is my great honour to be her PhD student. I appreciate her leading me to the area of finite permutation groups and symmetric graphs. I am also grateful for her suggestions in my mathematics writing. I am especially grateful to Professor Cai Heng Li who, as my co-supervisor, assisted me in both study and life from the time I arrived in Perth. His valuable discussions about my study will be of benefit to my career as a researcher in the future. My appreciation also extends to Dr. Jian Ping Wu, for all her kind help and friendship. My sincere thanks also extend to Dr. Alice Devillers who, as my co-supervisor, greatly contributed to my study. She encouraged me to learn the powerful MAGMA. She taught me how to write a research paper, and always corrected my writing mistakes carefully. I am also grateful to Dr. Michael Giudici, who provided a great deal of help at the commencement of my study. I appreciate greatly Professor Gordon Royle and Dr. John Bamberg, for their inspirational lectures in geometry and graph theory. I would also like to thank Winthrop Professor Ákos Seress, for his helpful discussion with the transitivity on 2-geodesics. My thanks go to Professors Wei Jun Liu and Shang Jin Xu, for their help in the early stages of my study. I gratefully acknowledge the financial support of the Scholarships for International Research Fees (SIRF) from the University of Western Australia. Finally, I dedicate this thesis to my parents for all their love and encouragement. I thank their support and understanding. 7

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9 Publications arising from this thesis 1. A. Devillers, W. Jin, C. H. Li and C. E. Praeger, Line graphs and geodesic transitivity, Ars Math. Contemp. 6 (2013), (Appears in Chapter 6) 2. A. Devillers, W. Jin, C. H. Li and C. E. Praeger, Local 2-geodesic transitivity and clique graphs, J. Combin. Theory Ser. A, 120 (2013), (Appears in Chapter 3) 3. A. Devillers, W. Jin, C. H. Li and C. E. Praeger, Finite 2-geodesic transitive graphs of prime valency, submitted. (Appears in Chapter 5) The author of this thesis was in charge of writing the above three papers, and contributed fully to the research plan, analysis, constructions and proofs, and handled the submissions. 4. A. Devillers, W. Jin, C. H. Li and C. E. Praeger, Finite normal 2-geodesic transitive Cayley graphs, in preparation. (Appears in Chapter 7) 5. A. Devillers, W. Jin, C. H. Li and C. E. Praeger, Comparison of distance, geodesic and arc transitivity for finite graphs, in preparation. (Appears in Chapter 4) The author of this thesis is also in charge of writing papers 4 and 5, and contributes fully to the research plan, analysis, constructions and proofs. He will handle the submissions. 9

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11 List of key symbols Groups G H G H G N G (H) C G (H) soc(g) α G G α Sym(n), S n A n Z n K : H Groups H is a subgroup of G H is a normal subgroup of G Normalizer of H in G where H G Centralizer of H in G where H G Socle of G G-orbit containing α Stabiliser of α in G Symmetric group of degree n Alternating group of degree n Cyclic group of order n Split extension of K by H 11

12 12 CONTENTS Graphs Γ, Σ Graphs V (Γ) Vertex set of Γ E(Γ) Edge set of Γ Aut(Γ) Automorphism group of Γ d Γ (u, v) Distance between a vertex u and a vertex v in Γ Γ(u) Neighbourhood of the vertex u in Γ Γ i (u) The set {v d Γ (u, v) = i, v V (Γ)} C n K n K m[b] diam(γ) girth(γ) val(γ) Cycle graph with n vertices Complete graph with n vertices Complete multipartite graph with m parts of size b Diameter of Γ Girth of Γ Valency of Γ

13 Chapter 1 Introduction The study of symmetry in graphs is one of the main themes in algebraic graph theory. By definition, a graph Γ is symmetric if its automorphism group Aut(Γ) is transitive on the set of ordered pairs of adjacent vertices. This study goes back to the classical work of Tutte [111, 112] in 1947 and 1959 showing that there are no 6-arc transitive graphs of valency 3. Since then, symmetry properties of graphs have attracted a great deal of attention and many important results have been proved, see a literature review in Section 1.2. This thesis is dedicated to the study of one particular symmetry property of graphs, namely s-geodesic transitivity. In differential geometry and physics, and indeed in common usage, a geodesic is a shortest line between two points on a curved or plane surface. In a discrete setting (for example a graph), there is a discrete notion of distance that allows us to define geodesics analogously. We define a geodesic from a vertex u to a vertex v of a graph Γ as any shortest path from u to v, where we measure distance by the number of edges. We call a geodesic an s-geodesic if the distance between u and v is s. In the infinite setting geodesics play an important role, for example, in the classification of infinite distance transitive graphs [77], and in studying locally finite graphs, see for example, [110]. In this thesis, we are interested in transitivity properties on geodesics in finite graphs. We say that a graph Γ is s-geodesic transitive if, it contains an s-geodesic, and for each i s, Aut(Γ) is transitive on the set of i-geodesics of Γ. Moreover, Γ is said to be geodesic transitive if it is s-geodesic transitive for all possible s. Many well-known finite graphs have this transitivity property. For instance, both the complete graph K n and the complete bipartite graph K n,n are geodesic transitive for any n 1. These two graphs have diameter at most 2. In Chapter 4, we will show that the 13

14 14 1. introduction Johnson graphs, Hamming graphs and Odd graphs are all geodesic transitive, and these graph families have both diameter and valency unbounded. 1.1 The aims and main results of the thesis Two well-known families of symmetric graphs, namely s-arc transitive graphs and s-distance transitive graphs, have a close relationship with s-geodesic transitive graphs: the first family has a stronger transitivity property and the other family has a weaker such property. We define s-arcs and s-arc transitive graphs firstly. For a positive integer s, an s-arc of Γ is an ordered (s + 1)-tuple (v 0, v 1,..., v s ) of vertices such that v i 1 and v i are adjacent for 1 i s, and v j 1 v j+1 for 1 j s 1. In particular, each 1-arc is a 1-geodesic, and each s-geodesic is an s-arc. The graph Γ is said to be s-arc transitive, if Aut(Γ) is transitive on the set of s-arcs. We usually refer to 1-arcs as arcs. Thus a 1-arc transitive graph is called an arc transitive graph, and also known as a symmetric graph. In particular, for each s 1, s-geodesic transitive graphs are symmetric. Since the class of symmetric graphs is so large, we turn our attention to s-geodesic transitive graphs with s 2. By definition, every s-geodesic is an s-arc, but some s-arcs may not be s-geodesics. If Γ has girth 3 (length of the shortest cycle is 3), then the 2-arcs contained in 3-cycles are not 2-geodesics. The only symmetric graph of girth 3 and valency 3 is the complete graph K 4 which contains no 2-geodesics. Thus 2-geodesic transitive graphs which are not 2-arc transitive have valency at least 4. The graph in Figure 1.1 is the octahedron which is 2-geodesic transitive but not 2-arc transitive and has valency 4. Thus the family of non-complete 2-arc transitive graphs is a proper subset of the family of 2-geodesic transitive graphs. If a non-complete graph Γ is 2-arc transitive, then the subgraph [Γ(u)] induced on Γ(u) is an empty graph (that is, with no edges), for each vertex u. However, there are many 2-geodesic transitive graphs whose neighbourhood subgraphs are not empty graphs. A graph is locally s-geodesic transitive if for each vertex u, the stabiliser of u is transitive on the i- geodesics starting from u for each i s, see Definition 1 of Subsection The first aim of this thesis is to determine the possible structures of [Γ(u)] for connected locally 2-geodesic transitive graphs. A 2-geodesic transitive graph is both locally 2-geodesic transitive and vertex transitive.

15 1.1. the aims and main results of the thesis 15 Figure 1.1: Octahedron 1 Theorem Let Γ be a connected non-complete locally (G, 2)-geodesic transitive graph such that each vertex has valency at least 2. Let u be a vertex of Γ. Then one of the following holds. (1) Γ is G-vertex transitive, girth(γ) = 3, [Γ(u)] is connected of diameter 2, and the induced action of G u on Γ(u) is transitive on vertices and on pairs of nonadjacent vertices. (2) Γ is G-vertex transitive, [Γ(u)] = mk r for some integers m 2, r 1. (3) Γ is not G-vertex transitive; Γ is bipartite with parts 1 and 2, and there exist m 1, m 2 2 such that for any u i i, [Γ(u i )] = m i K 1. Thus, if Γ is locally 2-geodesic transitive but not vertex transitive, then each 2-arc is a 2-geodesic, and so Γ is a locally 2-arc transitive graph. This family of graphs has been studied extensively. In this thesis, we are interested in locally 2- geodesic transitive graphs which are not locally 2-arc transitive, and so they are vertex transitive of girth 3. Following the above fundamental theorem, we investigate the locally disconnected case further. For two positive integers m, r, the family of locally mk r graphs is denoted by F(m, r) (see Definition 3.3.1), and for a graph Γ F(m, r), the clique graph of Γ is denoted by C(Γ) (see Subsection 2.4.1). Partial linear space and its point graph are defined in Section 2.5. Then we have the following theorem. Theorem Let Γ be a connected non-complete graph and m 2, r 1 be integers. Then the following two statements hold. (1) Γ F(m, r) if and only if Γ is the S-point graph of a partial linear space S of order (m, r + 1) which contains no triangles. (2) If Γ F(m, r), then the map φ m,r : Γ C(Γ) is a bijection from F(m, r) 1 The first two numbers stand for the chapter and the section, respectively, and the third number stands for the order of the theorem.

16 16 1. introduction to F(r + 1, m 1) and φ 1 m,r = φ r+1,m 1 in the sense that φ r+1,m 1 (φ m,r (Γ)) = Γ. In particular, C(C(Γ)) = Γ. Let Γ F(m, r) with m 2, r 1. We define S(Γ) = (P, L, I) to be the incidence structure with P = V (Γ), L = V (C(Γ)), and (p, l) I P L if and only if p l. Let S(Γ) be the incidence graph of S(Γ), see Definition Then Lemma shows that S(Γ) is the partial linear space corresponding to Γ of Theorem (1), and S(Γ) is its incidence graph. The above theorem shows that each Γ F(m, r) corresponds to the S(Γ)-point graph. Next, we will study the relation of transitivity properties between Γ and S(Γ) (see definitions of point- (G, s)-arc transitivity and line-(g, s)-arc transitivity in Subsection 3.3.2). A graph Γ is said to be locally s-arc transitive, if Γ contains an s-arc, and for any two s-arcs α and β beginning from the same vertex u, there exists g A u such that α g = β where A = Aut(Γ). We have the following theorem. Theorem Let Γ F(m, r) with m 2, r 1 and let G Aut(Γ). Then the following statements are equivalent. (1) Γ is (G, 2)-geodesic transitive. (2) S(Γ) is point-(g, 4)-arc transitive and locally (G, 3)-arc transitive. (3) S(Γ) is point-(g, 4)-arc transitive and line-(g, 1)-arc transitive. If s is less than or equal to the diameter of Γ, we can define another symmetry property for Γ, namely s-distance transitivity. We say that Γ is s-distance transitive if, for any two pairs of vertices (u 1, v 1 ), (u 2, v 2 ) with the same distance t s, there exists a graph automorphism g such that (u 1, v 1 ) g = (u 2, v 2 ). If Γ is s-distance transitive with s equal to the diameter of Γ, then Γ is said to be distance transitive, see a literature review in the next section. Thus, for a given s, the family of s- distance transitive graphs contains the family of s-geodesic transitive graphs, which in turn contains the family of s-arc transitive graphs. One purpose of the thesis is to investigate differences among the three transitivity properties. We know that if Γ has girth at least 2s + 1, then Γ is s-geodesic transitive whenever it is s-distance transitive, see Lemma However, this may not hold for the girth of Γ less than or equal to 2s. Thus, we propose to find graphs which are s-distance transitive but not s-geodesic transitive with girth at most 2s, for a given s 2. In Chapter 4, we will exhibit two infinite families of 2-distance transitive graphs

17 1.1. the aims and main results of the thesis 17 of diameter 2 and girth 3 which are not 2-geodesic transitive, namely the Paley graphs and Peisert graphs, see Theorem For s 3, we did not have examples at the time of writing this thesis. As we pointed out previously, we are particularly interested in 2-geodesic transitive graphs of girth 3, that is, which are not 2-arc transitive. We characterize those graphs of prime valency and valency 4. Let p be a prime such that p 1 (mod 4). Then we define the set C(p) in Definition In Theorem 5.1.2, we prove that a graph Γ C(p) if and only if Γ is a connected non-bipartite antipodal double cover of K p+1 with p 1 (mod 4); and for a given p, all graphs in C(p) are isomorphic, geodesic transitive but not 2-arc transitive and have diameter 3. In particular, the following theorem shows that the graphs in C(p) are the only 2-geodesic transitive graphs of prime valency that are not 2-arc transitive. Theorem Let Γ be a connected non-complete graph of prime valency p. Then Γ is 2-geodesic transitive if and only if Γ is 2-arc transitive, or p 1 (mod 4) and Γ C(p). The line graph L(Γ) of a graph Γ is the graph whose vertices are the edges of Γ, with two edges adjacent in L(Γ) if they have a vertex in common in Γ. We are interested in the relationship between the s-arc transitivity of a graph and the s-geodesic transitivity of its line graph. This is Theorem Theorem Let Γ be a connected regular, non-complete graph of girth g and valency at least 3. Let G Aut(Γ) and let s be a positive integer such that 2 s diam(l(γ)) + 1. Then Γ is (G, s)-arc transitive if and only if s g/2 + 1 and L(Γ) is (G, s 1)-geodesic transitive. As applications of Theorem 6.1.1, we first classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive, see Theorem Secondly we prove that the only non-complete locally cyclic 2- geodesic transitive graphs are the octahedron and the icosahedron, see Corollary Theorem Let Γ be a connected non-complete graph of valency 4. Then Γ is 2-geodesic transitive if and only if Γ is one of the following:

18 18 1. introduction (1) Γ is 2-arc transitive; (2) Γ = L(K 4 ); (3) Γ = L(Σ) for a connected 3-arc transitive cubic graph Σ. Moreover, Γ is geodesic transitive of girth 3 if and only if Γ = L(Σ) for a cubic distance transitive graph Σ, namely Σ = K 4, K 3,3, the Petersen graph, the Heawood graph or Tutte s 8-cage. Studying certain quotient graphs has been very successful in investigating many families of graphs, for example distance transitive graphs [100], s-arc transitive graphs [88], locally s-distance transitive graphs [29] and locally s-arc transitive graphs [47]. Thus in this research, it is very reasonable to study normal quotients to get a reduction theorem for the family of s-geodesic transitive graphs. We will prove reduction theorems for the family of s-geodesic transitive graphs with [Γ(u)] disconnected (Theorem 3.1.4), and the family in which [Γ(u)] is connected (Theorem 3.1.5). Then we study such basic graphs where G is quasiprimitive on the vertex set. This leads us to study normal (G, 2)-geodesic transitive Cayley graphs. The following theorem gives a general characterisation of this family of graphs. For an element a of a group T, o(a) denotes the order of a. Theorem Let Γ = Cay(T, S) be a normal (G, 2)-geodesic transitive Cayley graph. Then one of the following holds. (1) Γ = C r and T = Z r for some r 4. (2) Γ = K 4[2], T = Q 8 and S = T \ Z(T ). (3) There is a prime p and an integer q such that for all a S, o(a) = p and a \ {1} S, and o(b) = q for each b S 2 \ (S {1}). Finally, this family of graphs can be reduced too. Theorem Let Γ = Cay(T, S) be a normal (G, 2)-geodesic transitive Cayley graph. If T is not a minimal normal subgroup of G, then there exists a normal subgroup N of G such that N < T and one of the following holds. (1) Γ = K q f [p e ] for primes p, q and integers e, f, and Γ N = Kq f. (2) Γ is bipartite, (G, 2)-arc transitive and Γ N = K2. (3) Γ is a cover of Γ N = Cay(T/N, SN/N), where T/N is a minimal normal subgroup of G/N. Further, either Γ N = K S +1 is G/N-arc transitive, or Γ N is normal (G/N, 2)-geodesic transitive.

19 1.2. literature review 19 We will investigate further those graphs K q f [p e ] which are normal 2-geodesic transitive, see Theorem The graphs in Theorem (2) have been intensively studied as we mentioned above. Thus the further study of normal (G, 2)-geodesic transitive Cayley graphs reduces to the following three problems: investigating the case that T is a minimal normal subgroup of G, studying the 2-geodesic transitive covers of these graphs, and investigating the 2-geodesic transitive covers of complete graphs. 1.2 Literature review In this section, we review some of the literature about symmetry properties of graphs, including s-distance transitivity, local s-distance transitivity, s-arc transitivity and local s-arc transitivity. The study of finite distance-transitive graphs goes back to D. G. Higman [55] (in 1967), in which groups of maximal diameter were introduced. These are permutation groups which act distance transitively on some graph although the definition in [55] does not mention graphs. This study was followed by the investigation of some special classes of distance-transitive graphs. For example, Tutte cubic cages, the incidence graphs of the Desarguesian projective planes, the rank three graphs, etc. The theory and examples up to the 1970 s are reported in Biggs monograph [8, part 3]. Biggs and Smith gave a classification of valency 3 distance-transitive graphs in 1971, which is the first general result in the classification of distance-transitive graphs, see [10]. In the following twenty years, much progress was made on a classification of distance transitive graphs. For instance, Bannai, Ito et al. determined a bound on the diameter of distance transitive graphs in terms of the valency by a series of papers, see [4, 5, 16, 103, 108, 109, 115]. This success in diameter bounding of distance transitive graphs stimulated the classification of these graphs of small valency, and all distance transitive graphs of valency at most 13 are known, see [9, 39, 45, 46, 62, 101, 102]. For a connected graph Γ of diameter d we denote by Γ i (i = 1, 2,..., d) the graphs whose vertices are those of Γ and whose edges are the 2-subsets of points at mutual distance i in Γ. Then, Γ is said to be antipodal if its diameter d 2, and Γ d is a disjoint union of complete graphs. Now suppose that Γ is a finite distance transitive graph of valency greater than 2 and diameter greater than 1. In 1971, D.

20 20 1. introduction H. Smith proved that A := Aut(Γ) acts imprimitively on the vertices of Γ if and only if Γ is either bipartite or antipodal, see [100]. Suppose that Γ is bipartite with two parts P 1, P 2. We consider distance-2 graphs P 1 and P 2 of Γ. For i = 1, 2, we define V (P i ) = P i, and two vertices of P i are adjacent if and only if they have distance 2 in Γ. Smith [100] proved that P 1 and P 2 are isomorphic distance transitive graphs. Further, by Cameron [16, p.11], P i is not bipartite, and either P i is primitive or P i is antipodal and its antipodal quotient is primitive. On the other hand, if Γ is antipodal, then its antipodal quotient is neither antipodal nor bipartite, and so is primitive. Thus we obtain a distance-transitive graph which admits a primitive distance-transitive automorphism group, see also [15, p.141] or [89]. Hence in the classification project, attention is focused on the situation where A is primitive on the vertex set. In 1987, analysis of finite primitive distance transitive graphs using the O Nan- Scott Theorem was begun by Praeger, Saxl and Yokohama. The O Nan-Scott Theorem identifies eight types of primitive permutation groups, see [72, 97]. Praeger et al. [91] proved that only three types (affine type, almost simple type and product action type) can occur when A acts primitively on V (Γ). In the paper, they further showed that if A is primitive of product action type on the vertex set of Γ, then Γ is a Hamming graph or the complement of a Hamming graph of diameter 2. Thus they reduced the problem to the case where the automorphism group is either almost simple or affine. For more work on these two types, see [11, 12, 60, 71]. By a few decades of study, the classification of distance transitive graphs is almost complete except for some difficult cases, see the survey [13] by van Bon and the earlier survey [61] by Ivanov. Let s be a positive integer not exceeding the diameter of Γ. Then Γ is said to be locally s-distance transitive, if for each i s and each vertex u, the stabiliser A u is transitive on the set of vertices at distance i from u, where A = Aut(Γ). In particular, Γ is said to be locally distance transitive, if s is equal to the diameter of Γ. Thus s-distance transitive graphs are both locally s-distance transitive and vertex transitive. The family of s-distance transitive graphs are called s-symmetric graphs by Armanios in [1]. Assume that Γ is s-distance transitive of valency r 8 and = Ar or S r. If s 4, then Γ is known by [1, Theorem 1]. Armanios got more A Γ(u) u information for smaller valency r but not a complete classification. Suppose that Γ is not vertex transitive. Then these locally distance transitive graphs are called

21 1.2. literature review 21 distance-bitransitive graphs in [99] and are examples of distance biregular bipartite graphs studied by Delorme in [28]. A general study of locally s-distance transitive graphs was begun in [29] by Devillers, Giudici, Li and Praeger in In the paper they gave a nice reduction theorem for this family of graphs. Recall that a graph Γ is locally s-arc transitive, if Γ contains an s-arc, and for any two s-arcs α and β beginning from the same vertex u, there exists g A u such that α g = β where A = Aut(Γ). In particular, an s-arc transitive graph is both A-vertex transitive and locally s-arc transitive. As we pointed out previously, the first remarkable result about s-arc transitive graphs comes from Tutte who proved that for cubic graphs, there are no 6-arc transitive graphs, and the order of the stabiliser A u of a vertex u is at most 48. This seminal result stimulated greatly the study of s-arc transitive graphs. About twenty years later, relying on the classification of finite 2-transitive groups and thereby the finite simple groups classification, Weiss [114] proved that there were no 8-arc transitive graphs with valency at least three. These discoveries inspired people to construct s-arc transitive graphs for s {2, 3, 4, 5, 7}. Tutte himself in [111] had given a 5-arc transitive but not 6-arc transitive example, currently known as Tutte s 8-cage. In the following decades, many families of new 5-arc transitive graphs were constructed by Biggs and Conder, see [7, 20, 21]. Conder also constructed infinitely many 4-arc transitive graphs and 7-arc transitive graphs, see [22, 25]. In [88] (1993), Praeger gave a reduction theorem for the family of non-bipartite 2-arc transitive graphs. It showed that every non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. She also gave a new construction for a family of quasiprimitive 2-arc transitive graphs. In the same year, Ivanov and Praeger [63] classified vertexprimitive and vertex-biprimitive affine 2-arc transitive graphs. Eight years later, Li [70] determined all s-arc transitive graphs with s 4 for which A acts primitively or biprimitively on vertices. His classification involves the construction of new 4-arc transitive graphs, namely a graph of valency 14 admitting the Monster simple group M, and an infinite family of graphs of valency 5 admitting projective symplectic groups P Sp(4, p) with p prime and p ±1 (mod 8). Tutte s work also inspired the study of symmetric graphs under various restrictions, for example, on the valency, girth and order of the graph. For cubic symmetric graphs, R. C. Miller in [80] (1971) determined all such graphs with girth at most 6, and after many years of work, Foster [14] in 1988 published his census results on

22 22 1. introduction such graphs with up to 512 vertices. A few years later, a classification of 4-arc and 5-arc transitive cubic graphs with girth at most 13 was given (with some exceptions) by Morton in [81]. In 1995, with the help of a computer and based on an analysis of short relators in their automorphism groups, a complete classification of cubic symmetric graphs with order at most 240 was given by Conder and Morton [24], and they found one omission in the Foster census. A few years later, in [23], Conder and Dobcsányi listed all cubic symmetric graphs with up to 768 vertices. In the 1980 s, Lorimer [74, 75] investigated symmetric graphs with prime valency, Praeger and Xu [92] (1989) studied connected symmetric graphs of valency 2p whose automorphism groups have an abelian normal p-subgroup which is not semiregular on vertices, and in [93] (1993) classified vertex-primitive graphs of order a product of two distinct primes. A decade later, by analysing automorphism groups of graphs, Feng and Kwak [40] gave a classification of cubic symmetric graphs of order 2p 2, and using covering techniques, cubic symmetric graphs of order np or np 2 with 4 n 10 were classified in [41, 42, 43, 44]. Recently, Oh [82, 83] classified cubic symmetric graphs of order 14p or 16p. In all these studies p denotes a prime. If Γ is locally s-arc transitive but A := Aut(Γ) is not transitive on the vertex set, and if each vertex has valency at least 2, then Γ is edge transitive and bipartite, and A has two orbits on the vertex set. In 1978, Weiss proved that s 7 in the valency three case, see [113]. Stellmacher in 1996 in an as yet unpublished paper [104] announced that s 9 for locally s-arc transitive graphs with all vertices of valency at least three. In [47] (2003), Giudici, Li and Praeger analysed the families of finite locally primitive graphs and locally s-arc transitive graphs for s 2. They showed that taking quotients with respect to the orbits of a normal subgroup preserves both local primitivity and local s-arc transitivity and leads to the study of such graphs where A acts faithfully on both orbits and quasiprimitively on at least one. In the case where A acts quasiprimitively on both orbits, the possible quasiprimitive types were studied in [48], and it was shown that either the two quasiprimitive actions are of the same type, or one is of simple diagonal type and the other is of product action type, see Section 2.3. In [49], Giudici et al. studied further the case where the automorphism group acts quasiprimitively on only one bipartite half. Such graphs have a star normal quotient. For more work on s-arc transitive graphs and locally s-arc transitive graphs see [27, 38, 63, 68, 87, 88, 98]. The reduction method works very well in the study of symmetric graphs and

23 1.3. layout of thesis 23 locally symmetric graphs. This involves so-called quasiprimitive actions which were first defined and classified by Praeger in [88]. She divides finite quasiprimitive permutation groups into 8 types analogous to the O Nan-Scott types of finite primitive permutation groups (see Section 2.3). This has been a powerful tool for a lot of work on the study of symmetric or locally symmetric graphs, see for example [29, 37, 47]. 1.3 Layout of thesis Chapter 2. We give an overview of the basic definitions and some known results in finite groups, permutation groups, graphs and geometries, which will be used in the thesis. Chapter 3. This chapter plays a fundamental role in the investigation of (locally) 2-geodesic transitive graphs. In this chapter, we firstly determine the local structures of locally 2-geodesic transitive graphs in Theorem This theorem suggests a way forward for studying s-geodesic transitive graphs according to the structure of such graphs [Γ(u)] for a vertex u. We give a reduction for the family of s-geodesic transitive graphs where [Γ(u)] is connected and also a reduction for the family of s-geodesic transitive graphs where [Γ(u)] is disconnected. Next, we study further the family F(m, r) of connected graphs Γ such that [Γ(u)] = mk r for each vertex u, and for fixed m 2, r 1. We show that each Γ F(m, r) is the point graph of a partial linear space S of order (m, r + 1) which contains no triangles. Conversely, each S with these properties has point graph in F(m, r), and a natural duality on partial linear spaces induces a bijection F(m, r) F(r+1, m 1). The main results of this chapter appear in [31] (paper 2 on page 9). Chapter 4. In this chapter, we mainly compare s-geodesic transitivity of graphs with another two well-known transitivity properties, namely s-distance transitivity and s-arc transitivity. We first prove that the Hamming graphs, Johnson graphs and Odd graphs are geodesic transitive, these graphs have arbitrarily large diameter and valency, and so there is no upper bound s for s-geodesic transitive graphs, see Propositions 4.2.1, and Next, we prove that both the Paley graphs P (q) and the Peisert graphs P ei(q) are distance transitive but not geodesic transitive whenever q > 9 is a prime power, q 1 (mod 4) for P (q) and q = p e 1 (mod 4)

24 24 1. introduction where e is even and p 3 (mod 4) is a prime for P ei(q), see Theorem These results will appear in [34] (paper 5 on page 9). Chapter 5. This chapter is devoted to the classification of 2-geodesic transitive graphs of prime valency p which are not 2-arc transitive. We prove that p 1 (mod 4), and for each such p, there is a unique 2-geodesic transitive but not 2-arc transitive graph of valency p which is the antipodal double cover of the complete graph K p+1, see Theorems and 5.1.3, these two results appear in [32] (paper 3 on page 9). Chapter 6. In this chapter, we determine the relation between the s-arc transitivity of a graph Γ and the (s 1)-geodesic transitivity of its line graph L(Γ), see Theorem As an application, we classify 2-geodesic transitive but not 2-arc transitive graphs of valency 4, which is the smallest nontrivial such valency. We also determine which of these graphs are geodesic transitive. The main results of this chapter appear in [30] (paper 1 on page 9). Chapter 7. Suppose that Γ is a connected (G, 2)-geodesic transitive graph. By reduction results from Chapter 3, to study the basic such graphs, we can assume that G acts quasiprimitively on the vertex set V (Γ). Many of these basic graphs are Cayley graphs. This leads us to study (G, 2)-geodesic transitive Cayley graphs Cay(T, S) with G contained in the holomorph of T. This study can be further reduced to the following three problems: investigating the case that T is a minimal normal subgroup of G, studying the 2-geodesic transitive covers of these graphs, and investigating the 2-geodesic transitive covers of complete graphs. The main results of this chapter will appear in [33] (paper 4 on page 9).

25 Chapter 2 Notation, definitions and preliminary results In this chapter we collect some basic notation, definitions and some preliminary results relating to abstract groups, permutation groups, graphs and geometric structures that will be used in the subsequent chapters. 2.1 Finite group theory All groups in this thesis are finite groups, and the notation and terminology of group theory are standard, see for instance [94, 105, 106]. For convenience, we recall a few frequently used concepts and results. Let G be a group. A subgroup H of G is called a normal subgroup if H g = H for any g G, and we write H G. Clearly, the identity and G itself are two normal subgroups, called trivial normal subgroups. The group G is called a simple group if it has no nontrivial normal subgroups. An abelian group is a group G satisfying that ab = ba for all elements a, b of G. And G is said to be solvable if it has an abelian series, by which we mean a series 1 = G 0 < G 1 < < G n 1 < G n = G in which G r G r+1, and each factor group G r+1 /G r is abelian. Let p be a prime. Then a group G is called a p-group if the order of G is a power of p. A Sylow subgroup H of a group G is a subgroup of G such that H = p i where p is a prime number, p i G and ( H, G /p i ) = 1. This subgroup is usually called a Sylow p-subgroup. We state the well-known Sylow Theorem. 25

26 26 2. notation, definitions and preliminary results Theorem ([105, p.95]) Let G be a finite group and p be a prime number such that p G. Then there is a Sylow p-subgroup of G; any two Sylow p-subgroups of G are conjugate in G; and every p-subgroup of G is contained in a Sylow p-subgroup. The exponent of a finite group G is the least common multiple of the orders of all elements of G. The center of G is the subgroup {g G gx = xg, x G}, and is denoted by Z(G). For a subgroup H of G, the normaliser of H in G is the subgroup N G (H) = {g G H g = H}, and the centraliser of H in G is the subgroup C G (H) = {g G h g = h, h H}. 2.2 Permutation groups In this thesis we use standard notation and definitions for permutation groups, which can be found in [35, 117]. For a finite set Ω, a bijection from Ω to itself is called a permutation of Ω. The set of all permutations of Ω forms a group, under composition of mappings, called the symmetric group on Ω, denoted by Sym(Ω). Any subgroup of Sym(Ω) is called a permutation group of Ω. In particular, if the size of set Ω is n, then Sym(Ω) is denoted by Sym(n) or S n and any subgroup of Sym(n) is said to be of degree n. Let G be a group and Ω be a nonempty set. Assume that for every a Ω and every g G we have defined an element of Ω denoted by a g. Then we say that this defines an action of G on Ω if the following two conditions hold: (1) a 1 = a for all a Ω, where 1 is the identity element of G; (2) (a g ) h = a gh for all a Ω and all g, h Ω. In other words, an action of G on Ω is a mapping (a, x) a x from Ω G to Ω satisfying the above two conditions. The size Ω of Ω is called the degree of the G action on Ω. An element a of Ω is said to be fixed by the element g G if a g = a. The elements of G which fix all elements of Ω form a subgroup of G, called the kernel of this action. If the kernel is equal to the identity subgroup of G, then we say the action is faithful. Suppose a group G acts on a set Ω. For a point a Ω, we denote by a G = {a g g G} the set of images of a under G, and this set is called the orbit of a under G. The stabiliser of a in G is the following subgroup of G, G a = {g G a g = a}. The orbit of a under G and the stabiliser of a in G have the following property.

27 2.2. permutation groups 27 Lemma ([35, Theorem 1.4A]) Suppose that G acts on a set Ω and that g, h G and a, b Ω. Then (1) Two orbits a G and b G are either equal or disjoint. (2) G b = g 1 G a g whenever b = a g. (3) a G = G : G a. The group G is said to be transitive on Ω, if it has exactly one orbit, that is, a G = Ω for all a Ω. A transitive group G is said to be regular on Ω, if for any element a Ω, the stabiliser G a = 1; G is said to be 2-transitive on Ω, if G a is transitive on Ω \ {a}. Suppose that G acts on the set Ω. Then for each element g G there exists a mapping g of Ω into itself, given by a a g. Since g 1 is the inverse of g, it follows that the mapping g is a bijection. Thus we have a function ρ : G Sym(Ω) given by ρ(g) = g. Further, by the above conditions (1) and (2) it is easy to check that ρ is a group homomorphism. In general, any group homomorphism from group G into Sym(Ω) is called a permutation representation of G on Ω. Let ρ : G Sym(Ω) be a permutation representation of G on the set Ω. Define an action of G on Ω by setting a g = a ρ(g) for all a Ω and x G. Then ρ is the permutation representation which corresponds to this action. Thus we may think of group actions and permutation representations as different ways of describing the same situation. In the following we give some well-known permutation representations which will be used in the thesis. Example Let G be a finite group and let Ω = G. For each g G define two maps of G into itself as follows: R(g) : x xg, for all x G. L(g) : x g 1 x, for all x G. Then R(g) is called the right multiplication induced by g, and L(g) is called the left multiplication induced by g. Further, both R(G) and L(G) are subgroups of Sym(G) and are isomorphic to G, and both act regularly on G. The maps ρ 1 : g R(g) and ρ 2 : g L(g) give two permutation representations of G, called the right, left permutation representation, respectively. Example Let G be a finite group and H G. Let Ω = [G : H] be the set of right cosets of H in G. Let g G. Define

28 28 2. notation, definitions and preliminary results g : Hx Hxg, for all Hx Ω. Then g defines an action of G on Ω, and is called the coset action induced by g. The kernel of this action is g G g 1 Hg, which is called the core of H in G and is denoted by core G (H). If core G (H) = 1, then H is said to be core-free in G. Let M, K be groups and suppose that we have an action of M on K which respects the group structure of K; so for each x M the mapping u u x is an automorphism of K. Put G := {(u, x) u K, x M} and define a product on G by (u, x)(v, y) = (uv x 1, xy) for all (u, x), (v, y) G. Then G is a group called the semidirect product of K by M. 2.3 Primitive groups and quasiprimitive groups In this section, we present two families of permutation groups with strong transitivity properties. For notation and definitions refer to [35, 88, 89]. Let G be a group acting transitively on a set Ω. A partition of Ω is a set B = {B 1, B 2,..., B n } of non-empty subsets of Ω such that B i B j = whenever i j and Ω = B 1 B 2 B n. A partition B is G-invariant if for each g G, and each B i B, then B g i B. The partition into singletons and the partition with a single part are the trivial partitions of Ω, and all the other partitions are non-trivial. Each element B of a G-invariant partition is called a block of G. The whole set Ω is a block and the singleton {α} with α Ω are blocks, and they are called trivial blocks, and others are non-trivial blocks. The action of G on Ω is said to be primitive if G has no non-trivial G-invariant partitions; otherwise, the action is said to be imprimitive. Assume that G is imprimitive on Ω. Then there exists a non-trivial G-invariant partition of Ω, called an imprimitive partition or a system of imprimitivity of G. The following result gives two basic properties of primitive groups. Lemma ([35, Corollary 1.5A and Theorem 1.6A]) Let G be a group acting transitively on a set Ω with at least two points. Then the following hold. (1) G is primitive on Ω if and only if, the stabiliser G α is a maximal subgroup of G for every α Ω.

29 2.3. primitive groups and quasiprimitive groups 29 (2) If G is primitive on Ω, then every non-trivial normal subgroup of G is transitive on Ω. Note that the converse of Lemma (2) is not true. We call a transitive permutation group G on a set Ω quasiprimitive if all its non-trivial normal subgroups are transitive. Hence every primitive permutation group is a quasiprimitive group. There exist quasiprimitive but not primitive groups, for example, each transitive permutation representation of a nonabelian simple group for which a point stabiliser is not a maximal subgroup decides a quasiprimitive group which is not primitive. There is a remarkable classification of finite primitive permutation groups mainly due to M. O Nan and L. Scott, called the O Nan-Scott Theorem for primitive permutation groups, see [72, 97]. They independently gave a classification of finite primitive groups, and proposed their result at the Santa Cruz Conference in finite groups in The class of finite quasiprimitive permutation groups can be described in a fashion very similar to the description given by the O Nan-Scott Theorem for finite primitive permutation groups [88]. There are eight types of finite quasiprimitive permutation groups, which in most cases parallel the primitive types from the O Nan-Scott Theorem, and every finite quasiprimitive permutation group belongs to exactly one of these types [89]. For a finite group G, a nontrivial normal subgroup of G is called a minimal normal subgroup if it does not contain any other proper normal subgroup of G. The socle, soc(g), of G is the product of all minimal normal subgroups of G. The key to analysing finite quasiprimitive groups is to investigate their socles. The following theorem is originally stated for primitive groups, but it also applies to quasiprimitive groups. Lemma ([35, Theorem 4.3B] and [88]) Let G be a finite quasiprimitive group acting on a set Ω, and let K be a minimal normal subgroup of G. Then exactly one of the following holds. (1) For some prime number p and some integer d, K is a regular elementary abelian group of order p d, and soc(g) = K = C G (K). (2) K is a regular nonabelian characteristically simple group, C G (K) is a minimal normal subgroup of G which is permutation isomorphic to K, and soc(g) = K C G (K). (3) K is nonabelian, C G (K) = 1 and soc(g) = K.

30 30 2. notation, definitions and preliminary results The first three quasiprimitive types, namely types HA, HS, and HC, are exactly the same as the corresponding primitive types, that is, all quasiprimitive permutation groups of these types are primitive. For the HS type, the quasipimitive permutation group G has only one minimal normal subgroup; for types HS and HC, G has exactly two minimal normal subgroups, and these two minimal normal subgroups are isomorphic. (1) Holomorph affine type (HA): Ω = Z d p is an elementary abelian group, for a prime p and a positive integer d, and G is the semidirect product G = N : G α, a subgroup of the affine group AGL(d, p) on Ω, where N is the group of translations and G α is an irreducible subgroup of GL(d, p). The subgroup N is the unique minimal normal subgroup of G and is elementary abelian and regular on Ω. (2) Holomorph simple type (HS): Ω = T is a finite nonabelian simple group, and we have T.Inn(T ) G T.Aut(T ), where for α Ω, t T and σ Aut(T ), tσ : α α σ t σ. The group G has two minimal normal subgroups isomorphic to T. If α = 1 T, then Inn(T ) G α Aut(T ). (3) Holomorph compound type (HC): Ω = T k where k > 1 and T is a finite nonabelian simple group, and N = T k is a minimal normal subgroup of G which is regular on Ω. Further, NInn(N) G N.Aut(N), and G has two minimal normal subgroups isomorphic to N. If α = 1 T, then Inn(N) G α Aut(N), and G α acts transitively by conjugation on the simple direct factors of N. There are five other quasiprimitive types and for each of these types, H = soc(g) is the unique minimal normal subgroup of G and is nonabelian. (4) Almost simple type (AS): T G Aut(T ) where T is a finite nonabelian simple group, and G = T.G α. Here T is transitive and possible regular on Ω, that is, the stabiliser G α may be trivial. Note that, if G is a primitive permutation group of type AS, then its stabiliser G α must be a maximal subgroup of G, and in particular, G α {1 G }. The next two types are the diagonal types. (5) Simple diagonal type (SD): G is a subgroup of the group W = {(a 1, a 2,..., a k )φ a i Aut(T ), φ S k, k > 1, a i a j (mod Inn(T )), i, j}, where T is a nonabelian simple group, and Ω = T k 1. The action of W on Ω is the following: (a 1,..., a k ) : (t 1,..., t k 1 ) (a 1 k t 1a 1,..., a 1 k t k 1a k 1 ) and φ : (t 1,..., t k 1 ) (t 1 t k φ 1 1 φ 1,..., t 1 t k φ 1 (k 1) φ 1 ).

31 2.3. primitive groups and quasiprimitive groups 31 Thus for α = (1 T,..., 1 T ) Ω, W α = {(b,..., b)φ b Aut(T )} = Aut(T ) S k. The socle H = soc(g) = T k, and G acts transitively by conjugation on the k simple direct factors of H. Note that, if G is a primitive permutation group of type SD, then G acts primitively by conjugation on the k simple direct factors of H. (6) Compound diagonal type (CD): Ω = l and H = T k G N S l Sym( ) S l, for some divisor l of k and a nonabelian simple group T, where l 2 and k/l 2 and N Sym( ), soc(n) = T k/l and N is quasiprimitive of type SD. The group G acts transitively by conjugation on the simple direct factors of H. Note that, if G is a primitive permutation group of type CD, then N is primitive of type SD. The next type is the twisted wreath type. Here we follow the version in Suzuki s book [105, p.269]. Recall the notion of a core of a subgroup H. The twisted wreath product T twr φ P of groups T and P relative to φ is defined as follows. Let P have a transitive action on {1,..., k} and let Q be the stabiliser of the point 1 in the action. Suppose that there is a homomorphism φ : Q Aut(T ) such that core P (φ 1 (Inn(T ))) = {1 P }. Define N := {f : P T f(pq) = f(p) φ(q), p P, q Q}. Then N is a group under pointwise multiplication and N = T k. Let P act on N by f p (x) := f(px), p, x P. We define T twr φ P to be the semidirect product of N by P relative to this conjugation action of P. Such a twisted wreath product T twr φ P has a transitive action on N such that N acts regularly by right multiplication and for f N and p P, p : f f p. (7) Twisted wreath type (TW): G is the twisted wreath product T twr φ P and Ω = N with the action defined above. The differences between the primitive type TW and quasiprimitive type TW are subtle, see a discussion in [88, Remark 2.1]. (8) Product action type (PA): Ω = k, k > 1, N = T k G H S k Sym( ) S k where H is a quasiprimitive permutation group on of type AS with non-regular socle T, and G acts transitively by conjugation on the k simple direct factors of N.