关于Cayley图的若干研究

群的Cayley图构造简单和高度对称，成为群与图的一个重要的研究领域。Semi-Cayley图是Cayley图的自然推广。本项目围绕群的Cayley图和semi-Cayley图,探索刻画奇阶交换群的Cayley图的几乎2-可扩性和交换群的semi-Cayley图的1-和2-可扩性；研究群的Cayley图和semi-Cayley图的哈密尔顿圈(路)可扩性；研究交换群的semi-Cayley图的电阻距离和基尔霍夫指标，给出它的电阻距离和基尔霍夫指标公式；研究群的Cayley箭图的路代数及它的商代数，确定什么样的Artin代数是群的Cayley箭图的路代数的商代数。本项目的研究将丰富群与图的研究内容，不仅有重要的理论意义，且具有很好的应用背景。

Cayley graphs of groups with simple construction and high symmetry are an important research field in groups and graphs. Semi-Cayley graphs are natural generalization of Cayley graphs. This project is concerning with the Cayley graphs and semi-Cayley graphs of groups. In the project, we characterize the near 2-extendable Cayley graphs of abelian groups of odd order and the 1- and 2-extendable semi-Cayley graphs of abelian groups. We study the Hamilton cycle (path) extendability of Cayley graphs and semi-Cayley graphs of groups. We study the resistance distance and Kirchhoff index of semi-Cayley graphs of abelian groups, and give the formulae for its resistance distance and Kirchhoff index. We study the path algebra of a Cayley quiver and its quotient algebra, and consider what kind of Artin algebras are quotients of path algebras of Cayley quivers. The project will enrich the study of groups and graphs and have both theoretical and pratical values.