半弧传递双Cayley图

图的对称性是代数图论研究领域的一个前沿课题，也是群论在图论中的一个重要应用。在实际应用中，该研究与编码、密码、计算机网络设计等有着紧密联系。半弧传递图是图的对称性中一类重要的图，是由Tutte首先提出并研究的。之后，构造和分类半弧传递图受到国内外学者的广泛关注和研究。.本项目拟研究一类重要的高对称性图—双Cayley图。双Cayley图是Cayley图的自然推广，在图的对称性研究方面起着比较重要的作用。本项目将致力于研究半弧传递双Cayley图。主要研究内容包括：1.交换群、二面体群和非交换亚循环p-群上给定度数的半弧传递双Cayley图的分类或刻画；2.应用半弧传递双Cayley图的分类结果，研究二倍素数幂阶特定度数的半弧传递图的分类。主要目的是发展有关半弧传递双Cayley图的有用理论方法，在该方面一些难点问题上取得进展。

The symmetry of graphs is a front topic in algebraic graph theory. It is also an important application of group theory to graph theory. In practical applications, this research has closely linked with the code, password, computer network design,etc. Half-arc-transitive graphs are an important class graphs in the symmetry of garphs, which were first proposed and studied by Tutte. After that, the construction and classification of half-arc-transitive graphs have received extensive attention and research from scholars at home and abroad. .In this project, we will propose to study one important class symmetrical graphs, namely, bi-Cayley graphs. The bi-Cayley graphs are the natural generalization of Cayley graphs, which are playing an important role in the study of symmetry of graphs. The purpose of this project is to investigate half-arc-transitive bi-Cayley graphs. Specifically, the main researches are as follows: 1. The classification or characterization of given valency of half-arc-transitive bi-Cayley graphs over abelian group, dihedral group or non-abelian metacyclic p-group; 2. give the classification of certain valency half-arc-transitive graphs of order tiwce prime power based on the results of half-arc-transitive bi-Cayley graphs. The purpose of these studies is to develop useful theory for half-arc-transitive bi-Cayley graphs, and to make progress on some difficult problems in this area.