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 Gα of a vertex α acts primitively on the set (α) of vertices of r adjacent to α. For a finite non-abelian simple group L and a Cayley subset S of L, suppose that L 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 | Out( L)| , or L = PΩ8+ ( 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.