Prime-valent symmetric graphs with a quasi-semiregular automorphism

Abstract

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known problem regarding existence of semiregular automorphisms in vertex-transitive graphs. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers (Feng et al., 2019 [11]) and (Yin and Feng, 2021 [42]).Let Γ be a connected symmetric graph of prime valency p≥5 admitting a quasi-semiregular automorphism. In this paper, it is proved that either Γ is a connected Cayley graph Cay(M,S) such that M is a 2-group admitting a fixed-point-free automorphism of order p with S as an orbit of involutions, or Γ is a normal N-cover of a T-arc-transitive graph of valency p admitting a quasi-semiregular automorphism, where T is a non-abelian simple group and N is a nilpotent group. Further, for p=5 a complete classification of graphs Γ such that either Aut(Γ) has a solvable arc-transitive subgroup or Γ is T-arc-transitive with T a non-abelian simple group is given. Finally, a construction of an infinite family of symmetric graphs admitting a quasi-semiregular automorphism and having nonsolvable automorphism group is given.