A note on pentavalent -transitive graphs

A graph, with a group G of its automorphisms, is said to be ( G , s ) -transitive if G is transitive on s -arcs but not on ( s + 1 ) -arcs of the graph. Let X be a connected ( G , s ) -transitive graph for some s ≥ 1 , and let G v be the stabilizer of a vertex v ∈ V ( X ) in G . In this paper, we determine the structure of G v when X has valency 5 and G v is non-solvable. Together with the results of Zhou and Feng [J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010) 1725–1732], the structure of G v is completely determined when X has valency 5. For valency 3 or 4, the structure of G v is known.