S-Regular Graphs And Hamilton Graphs

The application of group theory to graph theory has been an important branch of mathematics, and symmetric graph and Hamilton graph are two important research problems in algebraic graph theory. In this thesis, we mainly investigate those two problems by using not only some theories and results of abstract groups, permutation groups and simple groups but also some methods and skills of graph theory and combinatorial theory.In Chapter 1, we mainly introduce some basic notations of group and graph. Also we introduce some relations between those notations. In the following, some related problems, reserch background knowledge and main results in the thesis are presented.In Chapter 2, we classify one-regular covering graphs of the complete bipartite graph K4,1, whose fibre-preserving group is arc-transitive and whose covering group is a cyclic group of prime order. As an application, we give a complete classification of tetravalent one-regular graph of order 8p, where p is a prime.In Chapter 3, we classify s-regular covering graphs of the complete bipartite graph K3,3, whose fibre-preserving group is arc-transitive and whose covering group is non-abelian groups of order p3 and Zp2×Zp. As an application, with the results of Feng's et al. in [J. Graph Theory,45 (2004) 101-112:J. Combin. Theory B,97 (2007) 627-646.], we classify the connected cubic symmetric graph of order 6p3.In Chapter 4, two sufficient conditions for vertex-transitive Hamilton graphs of prime-power order are obtained. With these conditions, two infinite families of non-Cayley vertex-transitive Hamilton graphs of order a 2-power are constructed.