On Graphs Admitting Arc-Transitive Actions Of Almost Simple Groups

A fundamental problem in determining the structure of a graphΓisthe problem of finding its full automorphism group Aut(Γ). This work ismainly to investigate the full automorphism group Aut(Γ) ofΓ, given itsalmost simple subgroup G.First, a new concept, namely, T-normal graph is introduced, whichis a natural generalization of the concept of normal Cayley graph. Thenwe consider T-vertex-transitive graphs with T a nonabelian simple groupand obtain a sufficient condition under which one can guarantee thatΓisa T-normal graph. Applying the result to G-arc-transitive graphsΓ, weprove that if the valency v(Γ) ofΓis at most 20 or a prime, thenΓisa soc(G)-normal graph for all but finite possibilities of soc(G). In par-ticular, if v(Γ) = 3, we eliminate all exceptions, that is, ifΓis a cubicG-arc-transitive graph thenΓis soc(G)-normal. By the T-normality, wedetermine the structure of the automorphism group Aut(Γ) ofΓ.Next, a construction is given for an infinite family of G-arc-transitivegraphs. Using this construction we give two family of quasiprimitve arc-transitive graphs which have non-quasiprimitive full automorphism groups.To the author's best knowledge, the only two infinite families of suchgraphs are constructed by Li [53] and Fang et al. [36], respectively.Finally, as an application of the above results, we completely deter-mine the automorphism group Aut(Γ) of cubic G-arc-transitive graphΓwith G = Ree(q)(q≥27), and, namely, we prove that Aut(Γ) = Ree(q) or Aut(Γ) = Ree(q)×Z₂. Moreover, we construct all cubic Ree(q)-arc-transitive graphs, amongst which exits there graphsΓsuch that Aut(Γ) =Ree(q)×Z₂.