On Tetravalent Vertex-transitive Bi-circulants

A graph ? is called a bi-Cayley if it admits a group H of automorphisins act-ing semiregularly on the vertices of ? with two orbits.Particularly.? is called bi-abelian or bi-circulant if H is abelian or cyclic group.This article will mainly focus on the bi-circulants with specific symmetric properties.A graph ? is said to be edge-transitiv(vertex-transitve or arc-tansitive)if the automorphism of ?acts transitively oil edge set(vertex set,arc set)of F.By now,a characterization of cubic edge-transitive bi-abelians has been classified(in particularly,all cubic edge-transitive bi-circulants are known).Recently,Kovacs et al classified all the trivalent and pentavalent bi-circulants.Motivated by these work,we aim to inves-tigate tetravalent vertex-transitive bi-circulants in this article.Let n be an odd integer,a complete classification is given of connected tetravalent vertex-transitive bi-circulants,which are non-Caylcy graphs.As by-products,a new infinite family of vertex-transitive non-Cayley graphs is constructed.As an application of the above results,this article also completely classified tetravalent vertex-transitive non-Cayley graphs of order 2p2(p is a prime).This article is organized as follows,Chapter 1 iutrocduces some basic defini-tions of group theory and graph theory;and suinmaries research questions and background knowledge which are relative to research contents of this articles.Chapter 2 gives some basic results of bi-Cayley graphs,and construct an infinite·family of connected tetravalent vertex-transitive non-Cayley bi-circuants.Chapter 3 investigates a complete classification of connected tetravalent vertex-transitive non-Cayley bi-Cayley graphs over cyclic group of order odd.Chapter 4 give,s a,com-plete classification of connected tetravalent vertex-transitive non-Cayley graphs of order twice a prime square.