2倍无平方因子阶的边传递图研究

Along with the development of the theory of finite permutation groups, the study about symmetry of various combinatorial structures by using permutation group theory has been becoming a very active field. In the present project we shall investigate, using group-theoretic and combinatorial methods, graphs with highly symmetric properties, and reveal the influence of the structure of automorphism group on the symmetric property of a graph; we also consider and try to settle several problems about permutation groups which are relative to the suborbits of some given finite permutation groups. This project emphasizes the study of edge-transitive graphs. We first investigate the structure of stabilizers of automorphism groups of edge-transitive graphs. By analyzing and computing suborbits of given permutation groups, we shall construct new families of edge-transitive graphs, and then, combining with combinatorial method, classify half transitive graphs and semisymmetric graphs with given orders or valencies.Theoretically speaking, every finite permutation group can be viewed as a subgroup of the automorphism group of an edge-transitive graph. Thus this project is not only important to the study on the symmetry of graphs but also.constructive to the theory of finite permutation groups.

随着有限置换群理论的不断完善，利用置换群理论考察组合结构的对称性已经成为国际学术界一个非常活跃的研究领域。本项目旨在结合群论和组合方法来考察具有较高对称性的图，揭示图的自同构群的结构对图的对称性的影响；同时，通过对某些图类的研究解决置换群理论中关于次轨道的某些问题。本项目将侧重于2倍无平方因子阶边传递图的研究，考察边传递图自同构群及其点稳定子的构造，通过分析、计算置换群的次轨道结构构造边传递图，在此基础上结合组合方法分类具有某些特定阶或度数的半传递图和半对称图。从理论上来说，每个有限置换群都可以是某个边传递图自同构群的子群。因此，无论是对图的对称性研究还是对置换群理论的完善，本项目都具有重要意义。