On 1-regular Dihedrants Of Valency Twice A Prime

The study of Cayley graphs is one of the most important topics in the field of symmetry of graphs.Cayley graphs play important roles in math-ematics and other subjects because of their simple structures and high symmetry properties.Let ? be a Cayley graph on a finite group G.If the regular subgroup R(G)is normal in the full automorphism group Aut(r)of ?,then ? is called a normal Cayley graph on G.and if Aut(?)acts regularly on its arcs,then ? is called 1-regular.In particular,a Cayley graph on a dihedral group is called a dihedrant.In this paper,we focus on 1-regular dihedrants of valency twice a prime.This thesis is organized as follows.Chapter 1 is an introduction part.We introduce some basic definitions of finite group theory and graph theory and the relative background of 1-regular dihecdrants.In chapter 2,we introduce some results related to dihedrants and the lexico-graphical product of graphs.In chapter 3,we give a complete classification of 1-regular normal dihedrants with dihedral vertex stabilizers.Combined with the results of Kwak et al.in[Journal of Combinatorial Theory,Series B,98(2008)585-598]and Wang et al.in[Acta Matheinatica Sinica,Chinese Series,3(2006)669-678],1-regular normal dihedrants of valency twice a prime have been determined.In chapter 4,we give a characterization of 1-regular non-normal dihedrants of valency six.Let ? be a connected 1-regular non-normal dihedrants of valency six.It is shown that there exists a semiregular cyclic subgroup H of Aut(?)such that the quotient graph ?H is isomorphic to the complete bipartite graph K6 6,the complete graph K4 of order 4,the cycle of length 4 or 6,or the lexicographical product K4[2K1].Moreover,the full automorphism group Aut(?)is depicted.Chapter 5 is a conclusion part.