On symmetric graphs of valency five

A graph XX, with a subgroup GG of the automorphism group Aut(X) of XX, is said to be (G,s)(G,s)-transitive  , for some s≥1s≥1, if GG is transitive on ss-arcs but not on (s+1)(s+1)-arcs, and ss-transitive   if it is (Aut(X),s)-transitive. Let XX be a connected (G,s)(G,s)-transitive   graph, and GvGv the stabilizer of a vertex v∈V(X)v∈V(X) in GG. If XX has valency 5 and GvGv is solvable, Weiss [R.M. Weiss, An application of pp-factorization methods to symmetric graphs, Math. Proc. Camb. Phil. Soc. 85 (1979) 43–48] proved that s≤3s≤3, and in this paper we prove that GvGv is isomorphic to the cyclic group Z5Z5, the dihedral group D10D10 or the dihedral group D20D20 for s=1s=1, the Frobenius group F20F20 or F20×Z2F20×Z2 for s=2s=2, or F20×Z4F20×Z4 for s=3s=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 GG, the automorphism group of Cay(G,S) is the semidirect product R(G)⋊Aut(G,S), where R(G)R(G) is the right regular representation of GG and Aut(G,S)={α∈Aut(G)∣Sα=S}.