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 Γ where G is an almost simple group with socle T . First we prove that, for an arbitrary subgroup M of Aut Γ containing G , either T is normal in M or T is a subgroup of the alternating group A k of degree k = | M α : T α | − | N M ( T ) : T | . Then we describe the structure of the full automorphism group of G -locally primitive graphs of valency d , where d ≤ 20 or is a prime. Finally, as one of the applications of our results, we determine the structure of the automorphism group Aut Γ for cubic symmetric graph Γ admitting a finite almost simple group.