(G, s)-Transitive Graphs of Valency 7

Let X be a finite simple undirected graph and G an automorphism group of X. If G is transitive on s-arcs but not on (s+1)-arcs then X is called (G,s)-transitive. Let X be a connected (G,s)-transitive graph of a prime valency p, and G(v) the vertex stabilizer of a vertex v is an element of V(X) in G. For the case p=3, the exact structure of G(v) has been determined by Djokovic and Miller in [Regular groups of automorphisms of cubic graphs, J. Combin. Theory (Ser. B) 29 (1980) 195 - 230]. For the case p=5, all the possibilities of G(v) have been given by Guo and Feng in [A note on pentavalent s-transitive graphs, Discrete Math.312 (2012) 2214 - 2216]. In this paper, we deal with the case p=7 and determine the exact structure of the vertex stabilizer G(v).