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

This paper investigates the automorphism group of a connected and undirected GG-symmetric graph ΓΓ where GG is an almost simple group with socle TT. First we prove that, for an arbitrary subgroup MM of AutΓ containing GG, either TT is normal in MM or TT is a subgroup of the alternating group AkAk of degree k=|Mα:Tα|−|NM(T):T|. Then we describe the structure of the full automorphism group of GG-locally primitive graphs of valency dd, where d≤20d≤20 or is a prime. Finally, as one of the applications of our results, we determine the structure of the automorphism group AutΓ for cubic symmetric graph ΓΓ admitting a finite almost simple group.