Two New Infinite Families Of Arc-regular Graphs

This dissertation is devoted to construct two new infinite families of arcregular graphs with cyclic and abelian vertex stabilizers respectively.The prove process also involves a classification of a family of edge-transitive Cayley graphs of dihedral groups which may have independent interests.A graph Γ is called arc-regular if its full automorphism group AutΓ acts regularly on its arc set.From the definition,arc-regular graphs are closely related to their full automorphism groups.As determining of the full automorphism groups of given graphs is one of the fundamental problems in the field of algebraic graph theory,constructing arc-regular graphs has received quite a lot of attention.The first interesting arc-regular cubic graph was constructed by Frucht [13].Quite a lot of known examples of arc-regular graphs are of small valencies(especially valency 3 and 4),see [6,10,12,28,29,32,33,47] for examples.Moreover,for prime valency,see Feng and Li [11],and for unbounded valency,see Kwak et at.[26,27].In this dissertation,two new infinite families of arc-regular graphs are given.Among them one family is consisted of arc-regular graphs with abelian(mostly not cyclic)vertex stabilizers.According to our knowledge,it is the first family with such a property in the literature.For proving the above results,we also give a classification of a family of edge-transitive Cayley graphs of dihedral groups.We note that 2-arc-transitive dihedrants have been classified by [9,30,31],and locally primitive dihedrants have been classified by [36].