边传递图与旗传递关联几何

The edge-transitive graph is an important research object of algebraic graph theory. In this project, we will study the flag-transitive incidence geometry using the methods of algebraic graph theory, and attack the isomorphism and classification problems of edge-transitive graphs by the incidence graphs of some important incidence geometry. The main research contents are given as follows: First, study the bitransitive permutation groups and edge-transitive graphs by the flag-transitive incidence geometry; Second, classificate certain important flag-transitive designs including the metacyclic and solvable Singer designs; Third, study the isomorphism problems of edge-transitive graphs, and try to given an criterion to the edge-transitive graphs. The researches of this project will expand the research contents of algebraic graph theory and promote the researches and developments of other fields of mathematics.

边传递图是代数图论的重要研究内容，本项目拟利用代数图论方法研究旗传递关联几何，并以重要的关联几何的关联图为导向，研究边传递图的同构问题和分类问题。具体研究内容如下：一、以旗传递关联几何为导向研究双侧传递置换群和边传递图；二、分类几类重要的旗传递设计，包括亚循环和可解Singer设计等；三、研究边传递图的同构问题，给出边传递图同构的判定准则。本项目研究将拓展代数图论的研究内容，推动相关数学领域的渗透研究和发展。

项目以传递图的同构问题及图的圈覆盖问题为主要研究内容，项目研究的重要结果包括：(1)针对点传递图的同构问题，对于给定群G，给出并利用G-点传递图为群G的GI-图的充分或者必要条件，构造了某些群G的GI-图，提出了GI-群和DGI-群的概念，探讨了二面体群等群类的DGI(GI)性质；(2)研究了边传递图的同构问题，利用图的自同构群的性质与子群结构，刻画了边传递图同构的判定准则；(3)刻画了广义Cayley图的同构问题，研究了某些特殊图类中的GCI-群，证明了交错群中只有A4是GCI-群，进一步研究了广义半Cayley图的同构问题；(4)研究了符号图的符号圈覆盖问题，改进了符号图的最短符号圈覆盖长度的上界; (5)充分考虑了图的极大偶因子，给出了极大偶因子长度的上界并刻画了极值图。