104 research outputs found

    On finite edge-primitive and edge-quasiprimitive graphs

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    Many famous graphs are edge-primitive, for example, the Heawood graph, the Tutte--Coxeter graph and the Higman--Sims graph. In this paper we systematically analyse edge-primitive and edge-quasiprimitive graphs via the O'Nan--Scott Theorem to determine the possible edge and vertex actions of such graphs. Many interesting examples are given and we also determine all GG-edge-primitive graphs for GG an almost simple group with socle PSL(2,q)PSL(2,q).Comment: 30 pages To appear in Journal of Combinatorial Theory Series

    Finite 33-connected homogeneous graphs

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    A finite graph \G is said to be {\em (G,3)(G,3)-((connected)) homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most 33 extends to an automorphism g∈Gg\in G of the graph, where GG is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite (G,3)(G, 3)-homogeneous graphs. In this paper, we develop a method for characterising (G,3)(G,3)-connected homogeneous graphs. It is shown that for a finite (G,3)(G,3)-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is 22--transitive or G_v^{\G(v)} is of rank 33 and \G has girth 33, and that the class of finite (G,3)(G,3)-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where GG is quasiprimitive on VV. We determine the possible quasiprimitive types for GG in this case and give new constructions of examples for some possible types

    Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph

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    In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a GG-vertex-transitive graph Γ\Gamma. In the main result the group GG is quasiprimitive or biquasiprimitive on the vertices of Γ\Gamma, and we obtain a genuine reduction to the case where GG is a nonabelian simple group. Using normal quotient techniques developed by the first author, the main theorem applies to general GG-vertex-transitive graphs which are GG-locally primitive (respectively, GG-locally quasiprimitive), that is, the stabiliser GαG_\alpha of a vertex α\alpha acts primitively (respectively quasiprimitively) on the set of vertices adjacent to α\alpha. We discuss how our results may be used to investigate conjectures by Richard Weiss (in 1978) and the first author (in 1998) that the order of GαG_\alpha is bounded above by some function depending only on the valency of Γ\Gamma, when Γ\Gamma is GG-locally primitive or GG-locally quasiprimitive, respectively
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