On the automorphism groups of symmetric graphs admitting an almost simple group

This paper investigates the automorphism group of a connected and undirected G-symmetric graph @C where G is an almost simple group with socle T. First we prove that, for an arbitrary subgroup M of [email protected] containing G, either T is normal in M or T is a subgroup of the alternating group A"k of degree k=|M"@a:T"@a|-|N"M(T):T|. Then we describe the structure of the full automorphism group of G-locally primitive graphs of valency d, where [email protected]?20 or is a prime. Finally, as one of the applications of our results, we determine the structure of the automorphism group [email protected] for cubic symmetric graph @C admitting a finite almost simple group.