(G,s)-Transitive Graphs of Valency 7

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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 Gv the vertex stabilizer of a vertex v ∈ V(X) in G. For the case p=3, the exact structure of Gv has been determined by Djoković 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 Gv 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 Gv.