A Family Of Edge-transitive Cayley Graphs Of Square-free Oder

This thesis mainly studies a family of edge-transitive Cayley graphs of odd square free order,and as consequences,two interesting characterizations regarding edge regular and arc regular are obtained.A positive integer is called square free if it has no factor being a square.The research of edge-transitive graphs of square free order has a rich history.Let p be a prime.In 1971,Chao[8]classified arc-transitive graphs of order p.Chen and Oxley[9]classified weak symmetric graphs of order 2p,and later Wang and Xu[35]determined all arc-transitive graphs of order 3p.This results were extended to the case of order a product of two distinct primes by joint papers of Praeger 1993.Moreover,edge-transitive graphs of square free order and valency at most seven were characterized in[11,19,21,23].However,it is challengable to give a nice characterization of general edge-transitive graphs of square free order.For example,a joint paper of Li[20]characterized the ’basic’ edge-transitive graphs(that is,each normal subgroup has at most two orbits on the vertex set)of square free order.Actually,the classification of edge-transitive graphs of order 6p has not been obtained yet.In this thesis,we obtain a complete classification of edge-transitive Cayley graphs of square free order and with valency less than the smallest prime divisor of the order of the graph.By using the obtained classification,the next two results are proved.For any given positive integers k,s ≥1 and m,n ≥ 2,there are infinitely many edge-regular normal metacirculants of valency 2m and order a product of n primes;such arc-regular and edge-regular examples are also specifically constructed.