On symmetric graphs of valency five

A graph X , with a subgroup G of the automorphism group Aut ( X ) of X , is said to be ( G , s ) -transitive, for some s ≥ 1 , if G is transitive on s -arcs but not on ( s + 1 ) -arcs, and s -transitive if it is ( Aut ( X ) , s ) -transitive. Let X be a connected ( G , s ) -transitive graph, and G v the stabilizer of a vertex v ∈ V ( X ) in G . If X has valency 5 and G v is solvable, Weiss [R.M. Weiss, An application of p -factorization methods to symmetric graphs, Math. Proc. Camb. Phil. Soc. 85 (1979) 43–48] proved that s ≤ 3 , and in this paper we prove that G v is isomorphic to the cyclic group Z 5 , the dihedral group D 10 or the dihedral group D 20 for s = 1 , the Frobenius group F 20 or F 20 × Z 2 for s = 2 , or F 20 × Z 4 for s = 3 . Furthermore, it is shown that for a connected 1-transitive Cayley graph Cay ( G , S ) of valency 5 on a non-abelian simple group G , the automorphism group of Cay ( G , S ) is the semidirect product R ( G ) ⋊ Aut ( G , S ) , where R ( G ) is the right regular representation of G and Aut ( G , S ) = { α ∈ Aut ( G ) ∣ S α = S } .