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THE VERTEX PRIMITIVE AND VERTEX BI-PRIMITIVE S-ARC REGULAR GRAPHS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Liang Niu, B.S., M.S. ***** The Ohio State University 2008 Dissertation Committee: Approved by Professor Akos´ Seress, Advisor Professor Ronald Solomon Advisor Professor Michael Davis Graduate Program in Mathematics ABSTRACT A complete classification is given of vertex primitive and vertex bi-primitive s-arc regular graphs with s ≥ 3. In particular, it is shown that the Petersen graph and Coxeter graph are the only vertex primitive 3-arc regular graphs, and that vertex bi-primitive 3-arc regular graphs consist of the complete bipartite graph K3,3, the standard double covers of the Petersen graph and Coxeter graph, and three graphs admitting PGL(2, 11), PGL(2, 13) and PΓL(2, 27), respectively. ii To My Family iii ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor Akos´ Seress for allowing me to work on this thesis. I have appreciated the many conversations we have had, both mathematical and otherwise, and your consistently warm and kind demeanor. You have always been quick to answer questions. It has been much appreciated. I would also like to thank my other committee members, Professor Ronald Solomon and Professor Michael Davis, for agreeing to serve on my committee. Professor Solomon gave me great help when I was working on this paper, and I would like to say that you were one of the very best professors I have had. You have been very giving of your time and gave great inspiration to me. I would also like to thank my former advisor, Professor Cai Heng Li. Although you have not been my advisor for a long time, you still gave me great advice on this paper. I highly appreciate your help. iv VITA 2003-Present . Graduate Teaching Associate, The Ohio State University 1999-2002 . Graduate Student, Peking University 2002 . M.S. in Mathematics, Peking University 1997 . B.S. in Mathematics, Hubei University FIELDS OF STUDY Major Field: Mathematics Specialization: Algebraic Graph theory v TABLE OF CONTENTS Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... v List of Figures . viii CHAPTER PAGE 1 Introduction and main results . 1 1.1 Basic background . 1 1.2 Main results . 5 2 Vertex stabilizers of s-arc regular graphs . 8 2.1 Theorem for the vertex stabilizers of s-arc regular graphs 8 2.2 The vertex stabilizers of s-arc regular graphs: s = 3 . 9 2.3 The vertex stabilizers of s-arc regular graphs: s ≥ 4 . 31 2.4 A more general result . 33 3 A key lemma . 37 3.1 The key lemma . 37 3.2 Some useful facts . 38 3.3 Proof of the key lemma, in the case p=2 . 43 3.4 Proof of the key lemma, in the case p odd . 45 4 Proof of the main theorems . 56 4.1 Basic background on coset graphs . 56 4.2 Proof of Theorem 1.2.1 and Corollary 1.2.2 . 58 vi 4.3 Proof of Theorem 1.2.3 and Corollary 1.2.4 . 63 Bibliography . 77 vii LIST OF FIGURES FIGURE PAGE 1.1 The Petersen Graph and Coxeter Graph . 3 viii CHAPTER 1 INTRODUCTION AND MAIN RESULTS 1.1 Basic background Denote by Γ an undirected connected graph with vertex set V and edge set E. For a vertex α ∈ V , denote by Γ (α) the neighborhood of α, that is the set of vertices adjacent to α in Γ . For a positive integer s, an s-arc of Γ is an (s + 1)-tuple (α0, α1, . , αs) of vertices such that αi ∈ Γ (αi−1) for 1 ≤ i ≤ s and αi−1 6= αi+1 for 1 ≤ i ≤ s − 1. A regular graph is a graph where each vertex has the same number of adjacent vertices, i.e. every vertex has the same valency. If Γ is regular of valency at least 3 and G ≤ Aut(Γ ) is transitive on the set of s-arcs of Γ , then Γ is called (G, s)-arc transitive. If G is regular on the set of s-arcs, then Γ is called (G, s)-arc regular. (Recall that a permutation group G is regular on a set Ω if G is transitive and |G| = |Ω|.) A graph Γ is called s-arc regular if it is (Aut(Γ ), s)-arc regular. The class of s-arc regular graphs is closely connected to some important classes of combinatorial objects, such as regular Mobius maps [16], near-polygonal graphs [15], and half-transitive graphs [18]. The purpose of this paper is to complete the classification of vertex primitive and vertex bi-primitive s-arc regular graphs with s ≥ 3. Recall that a graph Γ is called 1 vertex primitive if Aut(Γ ) is a primitive permutation group on the vertex set V . For a bipartite graph Γ with bi-parts ∆1 and ∆2, G ≤ Aut(Γ ) is called bi-primitive on + + the vertex set V of Γ if G is primitive on the bi-parts ∆i, i = 1, 2, where G is the setwise stabilizer of ∆1 (also ∆2) in G. A graph Γ is called vertex bi-primitive if Aut(Γ ) is bi-primitive on the vertex set V of Γ . Let Γ be a connected (G, s)-arc regular graph with vertex set V such that G is primitive or bi-primitive on V . If s ≥ 4, then (G, Γ ) can be read out from the main theorem of [14], which gives the classification of vertex primitive or vertex bi-primitive s-arc transitive graphs when s ≥ 4. If s = 2 and G is primitive on V then (G, Γ ) is classified in [6]. In this paper, we classify (G, Γ ) for s = 3. There are examples of (G, 3)-arc regular graphs with G primitive on the vertex set of the graph, such as the Petersen graph and Coxeter graph. First let us look at the Petersen graph, a regular graph of valency 3 with 10 vertices. The graph is ∼ ∼ the upper graph of Figure 1.1. It is well known that Aut(Γ ) = PGL(2, 5) = S5, and G = Aut(Γ ) is regular on the the set of 3-arcs. Further, G is primitive on the vertex ∼ set, since the vertex stabilizer Gα = D12 is maximal in G. Therefore the Petersen graph is a (G, 3)-arc regular graph with G = Aut(Γ ) primitive on the vertex set of the graph. Next let us look at the Coxeter graph, a regular graph of valency 3 with 28 vertices. The graph is the lower graph of Figure 1.1. We will prove later that Aut(Γ ) ∼= PGL(2, 7) and G = Aut(Γ ) is regular on the the set of 3-arcs. Further, we will ∼ see that G is primitive on the vertex set, since the vertex stabilizer Gα = D12 is 2 Figure 1.1: The Petersen Graph and Coxeter Graph 3 maximal in G. Therefore the Coxeter graph is another (G, 3)-arc regular graph with G = Aut(Γ ) primitive on the vertex set of the graph. In Corollary 1.2.2, we will see that the Petersen graph and the Coxeter graph are the only vertex primitive 3-arc regular graphs. A permutation group G ≤ Sym(Ω) is called sharply 2-transitive if it is 2-transitive on Ω and the identity is the only element of G with more than one fixed point. All sharply 2-transitive groups are listed below, see Theorem 9.4, chapter XII of [2]: Theorem 1.1.1. Assume that G is a sharply 2-transitive permutation group. Then d d G = Zp:Xω is affine, |Xω| = p − 1, and one of the following holds: d (I) Xω has a cyclic normal subgroup which is irreducible on Zp, and d Xω ≤ ΓL(1, p ) = Zpd−1.Zd; d d (II) Xω is solvable but has no irreducible cyclic normal subgroup on Zp, and (Xω, p ) is listed in the following table: d Xω p SL(2, 3) 52 2 G48 7 2 SL(2, 3) × Z5 11 2 G48 × Z11 23 where G48 = SL(2, 3).Z2 = Q8.S3 is the group defined in Definition 8.4, chapter XII of [2]. 4 d (III) Xω is not solvable, and (Xω, p ) is listed in the following table: d Xω p SL(2, 5) 112 2 SL(2, 5) × Z7 29 2 SL(2, 5) × Z29 59 1.2 Main results For the primitive case,we have: Theorem 1.2.1. Let Γ be a connected (G, 3)-arc regular graph such that G is prim- itive on the vertex set. Then either (i) Γ is the Petersen graph and G ∼= PGL(2, 5), or (ii) Γ is the Coxeter graph and G ∼= PGL(2, 7). And we have the corollary: Corollary 1.2.2. A connected graph Γ is vertex primitive 3-arc regular if and only if either (i) Γ is the Petersen graph and Aut(Γ ) ∼= PGL(2, 5), or (ii) Γ is the Coxeter graph and Aut(Γ ) ∼= PGL(2, 7). The above corollary shows that the Petersen graph and the Coxeter graph are the only vertex primitive 3-arc regular graphs. However, by [6, 14], there are infinitely many vertex primitive 2-arc regular graphs and 4-arc regular graphs. 5 For the bi-primitive case, we have: Theorem 1.2.3. Let Γ be a connected (G, 3)-arc regular graph such that G is bi- primitive on the vertex set.
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