无平方因子阶图的对称性和局部结构研究

This preject is a contribution to the application of group theory to graph theory. The application mainly investigates the different objects such as vertices, edges, arcs and so on. In the literature, the problem of characterizing and classifying edge-transitive and vertex-transitive graphs with square-free order has received considerable attentions. This problem for lots of special cases has been solved, many of which depended on the Liebeck-Saxl's classfication. During the postgraduate, the applicant obtained the structures of the automorphism group and its stabilizer subgroups and constructed a chine between them used two tools. One of the tools is Li-Seress's classification of primitive permutation groups with square-free degree given by Li and A. Seress in 2004,the other is the Thompson-Widlandt Theorem. Based on the structure of the chain, the applicant research 2-arc transitive graphs with square-free order in TianYuan. This project will continue to research the graphs with square-free order. The aim of our project is to characterize the locally-primitive graphs with certain valency, to give a complete classification of edge-transitive graphs with valency four, and to study the other symmetries of edge-transitive graphs,such as s-arc regular graphs. In the process of our study, there exists a class of graphs which is semiregular and has exactly two orbits on the vertices . Therefore, we can also extend edge-transitive graphs to Cayley graphs and Bi-Cayley graphs, also their properties. The research results of this project are meaning for the study of symmetries of graphs and the permutation group theory, also they can provide effective model in the application.

在代数图论中，图的对称性是一个重要的研究课题。近年来，刻划和分类无平方因子阶对称图已引起广泛关注。素因子个数较少时的图得到了刻划或分类，而对于一般的无平方因子阶对称图研究较少。利用2004年李才恒和A.Seress的无平方因子次数的本原置换群的分类和Thompson-Widlandt定理，申请人在读博期间得到无平方因子阶图对应的自同构群和其点稳定子群之间的一个子群链及该链含有的非可解合成因子结构。申请人2013年度的天元基金项目利用该结构刻画和分类无平方因子阶的2-弧传递图。本项目深化、扩展和完善上述已有的工作，继续研讨无平方因子阶图的对称性质。预期将分类和刻画无平方因子阶的局部本原弧传递图、本原图和s-弧正则图。此外，本项目还涉及无平方因子阶的双Cayley图。本项目的研究成果对图的对称性研究以及置换群理论的完善都有重要的意义。

本项目按照申报书中研究计划执行，主要是研究无平方因子阶图的对称性质和局部结构。具体为分类和刻画无平方因子阶的局部弧传递图、弧正则图和考察点稳定子群对对称性的影响。本项目刻画了自同构群的极大正规子群在无平方因子阶局部本原弧传递图顶点集至少有三个轨道时的图的结构。刻画了容许几乎单群和零散单群时的2-弧正则图。在考察点稳定子群对整个自同构群及对称性的影响时，发现稳定子群是否交换对整体结构有较大影响，进而着重研究元素的交换性质和广义交换性质。结合项目组成员研究方向和群中广义交换性质的探讨，对环中一些特殊性质加以研究，得到了具有某些特殊性质的环的刻画及其扩张性质。本项目采取项目组成员共同定期讨论、研究生上专业课程、外出交流学习等形式具体实施。基本完成了项目申报书中的预期研究结果，共发表4篇学术论文，投稿2篇，培养硕士研究生2名。