图的自同构群和对称性

In this project, we study the automorphism groups of graphs and investigate the symmetrical properties of graphs by considering the actions of groups on graphs, the structures of subgraphs and some related combinatorical structures, and we shall consider the following concrete problems. We shall inestigate graphs with stabilizers of given structures, determine the automorphism groups and symmetrical properties of these graphs and some classical combinatorical structures. We shall study the classification problem and reconstruction problem of locally pimitive graphs, in particular, classify or characterize that are metacyclic, graphs of odd order, graphs of almost simple type and graphs with given orders and valencies, etc.; we also investigate edge or arc regular subgroups of the automorphism groups of some locally primitive graphs,and consider their general embeddings. We shall extend our interesting to the class of semisymmetric graphs, characterize or classify some semisymmetric graphs of given girths, orders or valencies and semisymmetric graphs of affine type， investigate factorizations of semisymmetric graphs and semisymmetric factorizations of some of classes of graphs. Moreover, we shall investigate the incidence graphs of some combinatorical structures, finite geometries and etc., and construt some families of graphs with given symmetrical properties.

本项目主要研究图的自同构群的结构，通过考察群在图上的作用、图某些子图的结构和与之相关某些组合结构来研究图的对称性质，我们研究下面的一些具体问题。研究具有给定性质的点稳定子群的点或边传递图，确定图和某些经典组合结构的自同构群及其对称性质。研究某些局部本原图的分类及重构问题，主要涉及亚循环图、奇数阶图、几乎单型图以及给定点数或度数的图等；研究局部本原图自同构群的边或弧正则子群，进而考察某些图的正则嵌入。我们将研究兴趣延伸到半对称图，刻画和分类给定围长、阶或度数的半对称图、仿射型的半对称图、半对称图的因子分解以及某些图类的半对称图分解。此外，我们还将研究某些组合结构及有限几何等的关联图，借以构造新的具有给定对称性质的图类。

具有较高对称性的图，例如传递图、距离正则图等，有着良好的代数性质以及组合结构，它们是代数组合论的重要研究对象。这些图类本身蕴含着许多深刻而又困难的数学问题，比如自同构群的确定、局部结构与整体对称性的关系、图的存在性与分类等等。同时这些图类及相关理论在许多领域有着重要应用，例如置换群理论、单群及其子群结构、组合设计、极值组合等等。鉴于这些，本项目着重研究了下列具体问题：刻画给定传递性的图的自同构群，分析对称图的局部结构（包括自同构群的稳定子群）及其对对称性的影响，分类某些限定条件下的图类，研究某些图类相关的图论性质、Terwilliger代数及极值组合问题。在项目组成员努力工作和配合下，我们取得了一系列重要成果，发表或录用了45篇学术论文。