On the Weiss Conjecture for locally primitive graphs

A graph Gamma is said to be locally primitive if, for each vertex alpha, the stabilizer in Aut Gamma of alpha induces a primitive permutation group on the set of vertices adjacent to alpha. In 1978, Richard Weiss conjectured that for a finite vertex-transitive locally primitive graph Gamma, the number of automorphisms fixing a given vertex is bounded above by some function of the valency of Gamma. 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 Gamma contains a locally primitive subgroup that is almost simple.