Half-Transitive Graphs And One-Regular Graphs With Small Prine Valencies

The research of the symmetry of graphs has been a hot topic in groups and graphs theory. With the development of computer and net-work, the application of the symmetry of graphs has been increasingly more important in recent years, which draws many researchers’ atten-tion. The symmetry of graphs can be described by the transitive prop-erties of its automorphism group, which acts transitively on the vertex set, edge set and arc set.A graph is said to be half-transitive graph provided that its auto-morphism group is vertex-transitive and edge-transitive, but not arc-transitive. Recently, mang scholars are devoted to classifying and con-structing half-transitive graphs with small valencies, meanwhile there are some interesting results in their research. However, we have to face ac-tual difficulty in classifying and constructing half-transitive graphs with different valent cases. Therefore, it is worthy of being researched further on half-transitive graphs. A graph is defined as one-regular graph, if its automorphism group acts transitively on the vertex set, edge set and acts regularly on arc set. One-regular graph plays an important role in studying the symmetry of graphs.The content of the thesis goes around the half-transitive graphs and one-regular graphs. Firstly, the connected tetravalent half-transitive graphs of order qp2can be proved to be isomorphic to a normal Cayley graph, and the above graphs are isomorphic to some tightly attached graphs. For the automorphism of such graphs, its order and solvability can be determined. Secondly, the connected hexavalent half-transitive graphs of order3p2can be classfied partly, and the above graphs with hexavalent connect Cayley graph of a metacyclic group with order3p2can be constructed. Finally, one-regular Cayley of X(X is a subgroup of automorphism group) in case of5and7valency can be described with the help of the GAP software.