Quasi m-Cayley circulants

A graph ▫$Gamma$▫ is called a quasi ▫$m$▫-Cayley graph on a group ▫$G$▫ if there exists a vertex ▫$infty in V(Gamma)$▫ and a subgroup ▫$G$▫ of the vertex stabilizer ▫$text{Aut}(Gamma)_infty$▫ of the vertex ▫$infty$▫ in the full automorphism group ▫$text{Aut}(Gamma)$▫ of ▫$Gamma$▫, such that ▫$G$▫ acts semiregularly on ▫$V(Gamma) setminus {infty}$▫ with ▫$m$▫ orbits. If the vertex ▫$infty$▫ is adjacent to only one orbit of ▫$G$▫ on ▫$V(Gamma) setminus {infty}$▫, then ▫$Gamma$▫ is called a strongly quasi ▫$m$▫-Cayley graph on ▫$G$▫ .In this paper complete classifications of quasi 2-Cayley, quasi 3-Cayley and strongly quasi 4-Cayley connected circulants are given

Distance-transitive graphs admit semiregular automorphisms

A distance-transitive graph is a graph in which for every two ordered pairs ofvertices ▫$(u,v)$▫ and ▫$(u\u27,v\u27)$▫ such that the distance between ▫$u$▫ and ▫$v$▫ is equal to the distance between ▫$u\u27$▫ and ▫$v\u27$▫ there exists an automorphism of the graph mapping ▫$u$▫ to ▫$u\u27$▫ and ▫$v$▫ to ▫$v\u27$▫. A semiregular element of a permutation group is anon-identity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism