Researches And Constructions Of Several Families Of Transitive Graphs

This doctoral thesis is devoted to studying certain families of transitive graphs,involving characterizations and constructions of the graphs.Studying transitive graphs including vertex-transitive graphs,edge-transitive graphs and arc-transitive graphs(or symmetric graphs),which was initiated by a remarkable result of Tutte(1949)who proved that there is no finite s-arc-transitive graphs for an integer s≥ 6.A positive integer n is called square-free if it there is no prime p such that p2 divides n.Characterizing symmetric graphs of order a small number times a square-free integer is a hot,topic in the algebraic graph theory,and a series results for the valency at,most six have been obtained,but the results for the case of valency at least seven is rare.The first two main results of the thesis are giving complete classifications of 7-valent symmetric graphs of square-free order(see Theorem 1.2),and of order four times an odd square-free integer(see Theorem 1.3).The proofs also involve a classification of symmetric graphs of square-free order and any prime valency which admit,a soluble arc-transitive automorphism group.Transitive graphs with order twice a prime power provide a rich source of many interesting families of graphs,and have received considerable attention(also many characterizations have been applied for studying a lot of other families of transitive graphs as such graphs often appear as normal quotient graphs).However,to the best of our knowledge,there are rare results for the transitive graphs of order 2pn for the case’n ≥ 4’,where p is an odd prime.Since characterizing basic graphs(that is,the arc-transitive automorphism groups has no intransitive normal subgroup such that the graph is a normal cover of the corresponding normal quotient graph)is generally an important step towards the possible general classification,it would be an interesting topic to characterize basic symmetric graphs of order 2pn for general positive integer n.The one aim of this thesis is to solve this problem.Actually,we have obtained a complete classification of the basic symmetric graphs of order 2pn for any odd prime p and positive integer n,see Theorem 1.4.The study of self-complementary vertex-transitive graphs has a rich history.In 1962,Sachs constructed the first families of self-complementary circulants,and self-complementary vertex-transitive graphs have been used as models for finding lower bounds of Ramsey numbers.Recently,Li et al.found that the only insoluble com-position factor of the automorphism groups of self-complementary metacirculants is A5.In this thesis,we extending this result by proving the following interestingresults:(1).Any simple group is a section of automorphism groups of infinitely many self-complementary vertex-transitive graphs.(2).The automorphism groups of self-complementary vertex-transitive graphs of square-free order are soluble.(3).A5,A6 and PSL(2,7)are the only nonabelian simple sections of the au-tomorphism groups of self-complementary vertex-transitive graphs of 4-power-freeorder.