A characterisation on arc-transitive graphs of prime valency

Let Gamma be a finite simple undirected graph and G <= Aut(Gamma). If G is transitive on the set of s-arcs but not on the set of (s + 1)-arcs of Gamma, then Gamma is called (G, s)-transitive. For a connected (G, s)-transitive graph Gamma of prime valency, the vertex-stabilizer G(alpha) with alpha is an element of V (Gamma) has been determined when G(alpha) is solvable. In this paper, we give a characterization of the vertex-stabilizers of (G, s)-transitive graphs of prime valency when G(alpha) is unsolvable. (c) 2017 Elsevier Inc. All rights reserved.