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 AutΓ containing G, either T is normal in M or T is a subgroup of the alternating group Ak of degree k = |Mα : Tα| − |NM (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 applications of our results, we determine the structure of the automorphism group AutΓ for cubic symmetric graph Γ admitting a finite almost simple group.