| An ordered pair of two adjacent vertices of a graph is called an arc of the graph.An automorphism of a graph is a permutation x on the set of vertices such that{u,v} is an edge if and only of {ux,vx} is an edge.All automorphisms of a graph form a group,which is called the full automorphism group of the graph.A graph is said to be arc-transitive if its full automorphism group is transitive on the set of arcs.Arc-transitive graphs are also called symmetric graphs.Arc-transitive graphs are important objects in the field of algebraic graph theory and have been extensively studied.In this thesis,we study two classes of arc-transitive graphs.The first work is on quasi-semiregular arc-transitive graphs.A permutation is said to be quasi-semiregular if it is a product of one 1-cycle and some m-cycles in its distinct cycle decomposition with m>1.Arc-transitive graphs admitting quasi-semiregular automorphisms are called quasi-semiregular arc-transitive graphs for short.In 2019,Feng et al.studied quasi-semiregular arc-transitive graphs of valency three or four,but the case that the valency is four and the full automorphism group is insolvable is unsolved.First,we proved that there exists no such graph in the unsolved case.Next,we characterized quasi-semiregular arc-transitive graphs of prime valency,and it was shown that when the graph has a solvable arc-transitive automorphism group,the graph is Cayley graph on a 2-group with fixed-point-free automorphism,and when the graph has no solvable arctransitive automorphism group,such graph is a regular cover of some graph with almost simple full automorphism group.As an application,we further studied quasi-semiregular arc-transitive graphs of valency five,and prove that there are only three graphs with a solvable arc-transitive automorphism group,and there are only eight graphs with almost simple full automorphism group.Then,using the theory of coset graphs,for each prime p≠5,we constructed a quasi-semiregular arc-transitive graph with valency 5 and order 6p4,and determined its automorphism group.Furthermore,we proved that the constructed graph is the only quasi-semiregular arc-transitive graph of valency 5 and order 6p4.The second work is on arc-transitive Cayley graphs on nonabelian simple groups.For a Cayley graph Γ on a group G,the full automorphism group of Γ has a regular subgroup isomorphic to G,and if the regular subgroup is normal in the automorphism group of Γ,then Γ is called a normal Cayley graph on G,otherwise Γ is called a non-normal Cayley graph on G.In 2011,Fang et al.proposed a problem:classify non-normal locally primitive Cayley graphs on finite simple groups of valency d,where d≤20,or d is a prime number.This problem is completely solved only when d=3.Except this,a lot of efforts have been made to attack this problem by considering the following problem:characterize finite nonabelian simple groups admitting non-normal locally primitive Cayley graphs of valency d≥4.Even for this problem,it was only solved in the cases d≤5.and d=7 with the vertex stabilizer being solvable.We completely solving the second problem for the case when d≥11 is a prime and the vertex stabilizer is solvable,and it is proved that such graph is either among three sporadic Cayley graphs on A5 or M22,or a Cayley graphs on An,where n=pkl,and p is the valency of the graph,and k divides l,and k divides p-1,and k has the same parity as l.Furthermore,examples for those three sporadic Cayley graphs,and a family of Cayley graphs on Ap are given,where p≥5 is an odd prime. |
