Finite s -Arc Transitive Graphs of Prime-Power Order

Abstract

An s-arc in a graph is a vertex sequence (α0,α1,…,αs) such that {αi−1,αi} ∈ EΓ for 1 ⩽ i ⩽ s and αi−1 ≠ αi+1 for 1 ⩽ i ⩽ s − 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s + 1)-arcs. It is proved that if Γ is a finite connected s-transitive graph (where s ⩾ 2) of order a p-power with p prime, then s = 2 or 3; further, either s = 3 and Γ is a normal cover of the complete bipartite graph K 2 m , 2 m , or s = 2 and Γ is a normal cover of one of the following 2-transitive graphs: K p m + 1 (the complete graph of order pm + 1), K 2 m , 2 m − 2 m K 2 (the complete bipartite graph of order 2m + 1 minus a 1-factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2-arc transitive graphs were classified by Ivanov and Praeger in 1993.)