Vertex stabilizers of nite symmetric graphs and a strong version of the Sims conjecture

Well known Sims conjecture proved by P. J. Cameron, C. E. Praeger, J. Saxl and G. M. Seitz (1983) can be formulated as follows. Let be an undirected connected nite graph and G be a subgroup of Aut acting primitively on the vertex set V ( ) of . For x 2 V ( ) and positive integer i, denote by G x the point-wise stabilizer in G of the closed ball of the radius i with the center x of the graph . Then there exists a natural number c depending only on the valency of such that G x = 1. We obtain that G x = 1 for all and G from the Sims conjecture and, moreover, the constant 6 cannot be decreased. The proof, generalizations and corollaries of this result are discussed.