On the Weiss conjecture for finite locally primitive graphs

A graph Gamma is said to be locally primitive if, for each vertex alpha, th e 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 numb er 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 tru e for finite non-bipartite graphs provided that it is true in the case in w hich Aut Gamma contains a locally primitive subgroup that is almost simple.