Some Classes Of Edge-transitive Graphs

This thesis is devoted investigating several families of edge-transitive graphs,which belongs to the field of algebraic graphs theory.Especially,we studied the following two questions:normal edge-transitive graphs of order pq2,and half-arc-transitive graphs of order 2p3,where p and q are prlrime,respectively.Let ? = Cay(G,S)be a Cayley graph of group G with respect to S,and denote right regular representation of G by R(G).If R(G)is normal in Aut ? then ? is called normal Cayley graph,and if NAut?(R(G))acts transitively on E?.then r is called a normal edge-transitive graph.A graph is called normal half-arc-transitive if it automorphism group acts transitively on its vertex set,edge,but not arc set.The investigation of Cayley graph generally is divided into two steps,which are normal and non-normal.This thesis mainly considers normal edge-transitive Cayley graph.In 1999 Praeger C E firstly put up the concept of normal edge-transitive graph,and gave the necessary and sufficient conditions for normal edge-transitive Cayley graph.After that the normality of Cayley graphs has been extensively studied by many schoolars.For example,Darafsheh M R studied the normal edge-transitive Cayley graphs of order 4p in 2013.Praeger C E and Corr B P classified the normal edge-transitive Cayley graphs of order pq in 2015.These motivated us to consider the normal edge-transiuave Cayley graphs of order pq2.In chapter 3,we studied the normal edge-transitive Cayley graphs of order pq2,and proved that these graphs are all normal half-arc transitive.Construction of half-arc transitive is always one of the hot topics in algebraic graph field.Therefore,the construction of half-arc-transitive graph is an interesting work.For example,Xu M Y has classified the half-arc-transitive Cayley graphs of order ps in 1992.Wang X Y and Feng Y Q showed that there is no tetravalent half-arc transitive graphs,of order 2p2 in 2010.Hence,we investigated the tetravalent half-arc-transitive graphs of order 2p3 in this thesis.