Pentavalent Arc-transitive Graphs Admitting Quasi-semiregular Automorphisms

An automorphism of a graph is a permutation on the vertex set of the graph preserving the edge set of the graph.The set of all automorphisms of the graph,under the operation of composition,forms a group,called the automorphism group of the graph.A graph is called vertex-transitive or arc-transitive if the automor-phism group of the graph is transitive on the vertex set or the arc set of the graph,respectively.A permutation is called semiregular if its disjoint cycle decomposi-tion has the same length.Studying vertex-transitive graphs admitting semiregular automorphisms is a hot topic in algebraic graph theory.A permutation is called quasi-semiregular if its disjoint cycle decomposition contains a 1-cycle and other cycles have the same length greater than 1.Quasi-semiregular permutation is a generalization of semiregular permutation.Cubic and tetravalent arc-transitive graphs admitting quasi-semiregular automorphisms were classified in the litera-ture.In this dissertation,using voltage graph,we construct the first infinite family of pentavalent arc-transitive graphs admitting quasi-semiregular automorphisms,which is also a classification of elementary abelian coverings of the complete graph₆admitting quasi-semiregular automorphisms.