边本原图与弧传递图研究

The research of edge-primitive graphs and arc-transitive graphs is difficult, but hot issue in the symmetry of graphs, which is a major topic in algebraic graph theory and topological graph theory. This research is not only closely linked with other branches of mathematics, such as group theory, geometry, topology, but also has a wide range of applications in the field of networks, information science, cryptography, and molecular biology. Thus, this research has important theoretical significance and practical application value. In this project we will focus on the following three problems: 1. Characterization and classification of edge-primitive graphs. Using algebraic geometry, finite group theory and specially primitive group theory, and basing on the classification of cubic edge-primitive graphs given by R.M. Weiss in 1973, we study the further characterization and classification of edge-primitive graphs. 2. Vertex stabilizers of arc-transitive graphs. By using local analysis of arc-transitive graphs, finite group theory, projective geometry and combinatorics, we try to determine the exact structure of vertex stabilizers of such graphs. 3. Classification of arc-transitive graphs. As an application of the vertex stabilizers of arc-transitive graphs, we calssify some given calsses of such graphs by using finite group theory and quotient graph theory.

边本原图和弧传递图是图的对称性研究中的难点和热点，而图的对称性研究又是代数图论和拓扑图论中的重要研究分支。该方面的研究不仅与其他数学分支，如群论、几何学、拓扑学等有着密切的联系，而且在互联网络、信息科学、密码学以及分子生物学等领域也有着广泛的应用。因此该研究具有重要的理论意义和实际应用价值。本项目将围绕如下三个问题开展研究：1. 边本原图的刻画与分类。拟利用代数几何、有限群论、特别是本原群理论，在R.M. Weiss于1973年完成的三度边本原图的分类工作的基础上，进一步开展边本原图的刻画与分类工作。2. 研究弧传递图的点稳定子群的结构。拟通过对弧传递图的局部分析，利用有限群论、射影几何以及组合的方法确定弧传递图的点稳定子群的具体结构。3. 弧传递图的分类。作为对弧传递图的点稳定子群结构的应用，利用有限群理论、商图理论对给定图类进行分类。

本项目利用有限群理论、代数几何、组合图论等方法，并借助计算机软件包MAGMA研究边本原图与弧传递图，主要集中在小度数边本原图的分类；给定度数的弧传递图的点稳定子群的结构；小度数弧传递图的分类这三个问题上，并得到了如下成果：1. 4度边本原图的完全分类（发表在JCTB）；2. 6度、7度弧传递图的点稳定子群的结构；3. 素数度弧传递图的可解点稳定子群的结构；4. 利用点稳定子群的结构，给出了16p阶5度（p是素数）对称图，2pq^2阶(p,q是不同素数)3度半对称图的完全分类。上述的结果形成论文6篇，其中4篇发表，2篇录用。