S-Arc-Regular Graphs Of Prime Valency And Square-free Order

This dissertation is devoted to study s-arc-regular graphs of square-free order and prime valency with s ? 2.(Notice that a positive integer n is called square-free if there is no prime p,such that p2 | n.)A graph ? is called s-arc-regular,if its full automorphism group Aut ? is regular on its s-arc set.By definition,s-arc-regular graphs are closely related to their full automorphism groups.Since determining automorphisms of graphs generally difficult,and it's one of the fundamental topics in the field of algebraic graph theory,s-arc-regular has received considerable attention in the literature.See[11,12,13,15,19,32,47]for instance.In particular,Feng and Li determined all 1-arc-regular graphs of square-free order and prime valency in[14],and their result has been cited for more than fifty times in the literature.A graph ? is called Cayley graph of group G if there is a subset S(?)G\{1}such that S = S-1,the vertex set of ? is G and two vertex x and y are adjacent if and only if yx-1 ? S.The family of Cayley graphs is one of the most important families of graphs in the field of algebraic graph theory.According to[11],all 1-arc-regular graphs with prime valency are Cayley graphs,thus the result in[14]naturally motivated the following problem.Problem.Classifying s-arc-regular Cayley graphs of prime valency and square-free order.One of the main results of this dissertation is to completely solve this prob-lem.Also,for the non-Cayley graphs case,we have characterized 2-arc-regular graphs of square-free order and valency t with t ? 3(mod 4).The proving process also involves the classification of vertex stabilizers of s-arc-regular prime valent graphs,which will be useful for the study of relating problem.