边传递双Cayley图及其在图的对称性中的应用

图的对称性是群与图研究领域的前沿课题之一，在分子化学、生物学、计算机科学、网络科学等学科领域有着广泛应用。双Cayley图是一类具有较强对称性的图 (这类图有一个在其顶点集合上有两个轨道的半正则自同构群)。近年来，人们利用双Cayley图攻克了Godsil-2幂阶Cayley自同构群问题、Marusic-半对称四循环图问题等若干图的对称性方面的公开问题。随之，双Cayley图的对称性研究越来越多地引起了国内外学者的关注，日益成为一个新兴热点课题。本项目将围绕边传递双Cayley图开展如下研究：系统地研究循环群、交换群、二面体群、亚循环群、内交换群等基本群类上的边传递双Cayley图；发展一套研究边传递双Cayley图的理论，并应用其研究给定阶的边传递图、给定基图的边传递正则覆盖等方面的若干重点问题。其目的是解决这些课题中的一些关键难点问题，在每个课题上取得突破性进展。

Symmetry of graphs is one of the important research topics in the research field of groups and graphs, and it has been widely used in areas as diverse as molecular chemistry, biology, computer science and network science. Bi-Cayley graphs form a class of graphs with high levels of symmetry (these graphs admit a group of automorphisms acting semiregularly on its vertex set with two orbits). Recently, bi-Cayley graph has been used to solve many open problems in the field of graph symmetries such as Godsil’s problem on automorphism groups of Cayley graphs of order a 2-power, Marusic’s problem on semisymmetric tetracirculants and so on, and because of this, bi-Cayley graph has been drawn much attention of researchers all over the world, and is becoming one of the hot issues in the research field of graph symmetries. The purpose of this project is to consider the following topics regarding edge-transitive bi-Cayley graphs: Study edge-transitive bi-Cayley graphs over cyclic groups, abelian groups, dihedral groups, metacyclic groups, inner abelian groups etc.; Develop a theory of edge-transitive bi-Cayley graphs and use it to study edge-transtive graphs with given order, edge-transitive regular covers of graphs and so on. It aims to solve fundamental problems regarding bi-Cayley graphs, and to make significant and breaking progress.