Prime-valent symmetric graphs with a quasi-semiregular automorphism

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]).