Prime-valent symmetric graphs with a quasi-semiregular automorphism

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