On the weiss conjecture for finite locally primitive graphs

Abstract

A graph F is said to be locally primitive if, for each vertex a, the stabilizer in Autf of a induces a primitive permutation group on the set of vertices adjacent to a. In 1978, Richard Weiss conjectured that for a finite vertex-transitive locally primitive graph F, the number of automorphisms fixing a given vertex is bounded above by some function of the valency of F. In this paper we prove that the conjecture is true for finite non-bipartite graphs provided that it is true in the case in which Aut F contains a locally primitive subgroup that is almost simple.