On locally primitive Cayley graphs of finite simple groups

A graph /is said to be G-locally primitive, where G is a subgroup of automorphisms of /-, if the stabiliser G~ of a vertex a acts primitively on the s e t / ( a ) of vertices of / adjacent to or. For a finite non-abelian simple group L and a Cayiey subset S of L, suppose that L <3 G...<Aut( L ) , and the Cayley graph /= Cay ( L, S) is G-locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of /is an add prine divisor of I Ou t (L ) I, or L = PQ{ ( q ) and/ has valency 4. In either cases, it is proved that the full automorphism group of / is also almost simple with the same socle L.