Finite Dual Bi-Cayley Graphs And Normal Cayley Graphs

All groups considered in this thesis are finite,and all groups considered are finite,undirected and simple.Let Γ be a graph,and let Aut(Γ)denote its full automorphism group.Γ is called a Cayley graph if Aut(Γ)contains a subgroup which is regular on the vertices.Γ is called a bi-Cayley graph over a group H if Aut(Γ)contains a subgroup which is isomorphic to H and acts semi-regularly on the vertices with exactly two orbits.In particular,the bi-Cayley graph Γ is called a dual bi-Cayley graph if the left regular representation of H is a subgroup of Aut(Γ).In this thesis,we study the structures and properties of dual bi-Cayley graphs and normal arc-transitive Cayley graphs on generalized quaternion groups.In this thesis,necessary and sufficient conditions for a bi-Cayley graph to be a dual bi-Cayley graph and some special dual bi-Cayley graphs are given.By studying the standard double cover of dual bi-Cayley graphs and the normal quotient graph of dual Cayley graphs,we obtain that the standard double cover of a dual bi-Cayley graph is a dual bi-Cayley graph,and prove that the normal quotient graph of an edge-transitive dual bi-Cayley graph is an edge-transitive dual bi-Cayley graph.We also characterize the structure of semi-symmetric dual bi-Cayley graphs.By studying normal quotient graphs of normal arc-transitive Cayley graphs on generalized quaternion groups,the valency of these graghs is completely determined.Moreover,we completely classify such graphs for the case that the order is four times a prime and the valency is twice a prime and prove some properties of Cayley graphs on generalized quaternion groups.