| Graph theory is a branch of modern mathematics.Since its inception in the early20 th century,it has undergone long-term development and research.Cayley graphs are undirected graphs in graph theory,named after the British mathematician Arthur Cayley,one of the first people to systematically study group theory.The study of Cayley graphs was originally used to study the symmetry properties of algebraic groups and algebraic systems.The study of Cayley graphs has important applications in many fields of mathematics,such as group theory,graph theory,and computer science.In the study of graph theory,s-arc transitive graphs are a special type of graph that can reflect some special properties of graphs.The study of s-arc transitive graphs originated from Tutte’s(1949)result: a 3-degree graph is at most 5-arc transitive.Furthermore,Weiss proved that graphs with degree greater than 3 are at most 7-arc transitive.Classifying and characterizing s-arc transitive graphs with s ≤ 7 has always been one of the most popular problems in algebraic graph theory.Combining Cayley graphs with s-arc transitive graphs has promoted the development of algebraic graph theory.Specifically,s-arc transitive Cayley graphs can describe the operation of a group on a generating set,so they are important for understanding the interactions between elements in a group,finding subgroups,computing indices,and characterizing recursive groups.In addition,s-arc transitive Cayley graphs have also been applied in algebraic topology,algebraic geometry,and Lie groups,providing new ideas and tools for tree structure and graph algorithms in computer science.2-arc transitive graph is a relatively important class in s-arc transitive graphs.C.E.Praeger pointed out that any 2-arc transitive graph is either a primitive graph itself or can be represented as a normal covering of a 2-arc transitive basic graph.She proposed the open problem of classifying all 2-arc transitive basic graphs,which has attracted wide attention and made significant progress in recent years.In this work,we mainly study 2-arc transitive Cayley graphs on certain groups,where the groups we consider are Frobenius groups and the semidirect product of odd order commutative groups and cyclic groups of order 2.The structure of Frobenius groups and semidirect product groups is not uniquely determined.The groups studied in this thesis will contain commutative groups,and the classification of 2-arc transitive Cayley graphs on commutative groups has been given by Li and Pan.Therefore,we mainly consider non-commutative groups and list the structures of these groups in Chapter 3,and derive their normal subgroups from these properties.Based on this,we further study 2-arc transitive Cayley graphs on these groups and give the first classification results.Another result of this thesis is the classification of 2-arc transitive Cayley graphs on a more general class of given groups.For this problem,we characterize the 2-arc transitive Cayley basic graphs on the given group and prove that they are either bipartite or cover of complete graph. |
