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

In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a G-vertex-transitive graph @C. In the main result the group G is quasiprimitive or biquasiprimitive on the vertices of @C, and we obtain a genuine reduction to the case where G is a non-abelian simple group. Using normal quotient techniques developed by the first author, the main theorem applies to general G-vertex-transitive graphs which are G-locally primitive (respectively, G-locally quasiprimitive), that is, the stabiliser G"@a of a vertex @a acts primitively (respectively quasiprimitively) on the set of vertices adjacent to @a. 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 G"@a is bounded above by some function depending only on the valency of @C, when @C is G-locally primitive or G-locally quasiprimitive, respectively.