一类奇数阶2倍素数度的局部本原图的构造

Construction of a Class of Local Primitive Graphs of Odd Order and Twice Prime Valency

摘要: 早在1878年，A. Cayley为了解释群的生成元和定义关系提出了以他的名字命名的Cayley图。因为其构造的简便性和高度对称性，Cayley图得到了图论学者们的广泛重视。1938年，R. Frucht证明了对于任意给定的抽象群，都存在一个以这个群为自同构群的图。这项重要工作引发了一个新的研究领域——群与图，即以图的自同构群、群在图上的作用和具有各种传递性质的图作为主要的研究对象，同时也借助图来解决某些群论问题。在过去的80多年里，群与图已经发展成为代数图论的一个重要领域，先后出现了许多精彩的理论。本文是在奇数阶2倍素数度的条件下，借助于拟本原图的分类结果，通过分析图$\Gamma$ 的自同构群的子群及其点稳定子群的结构，利用本原群的分类结果，构造出了一类局部本原的拟本原图。

Abstract: As early as 1878, A. Cayley introduced the concept of the Cayley graph, named after him, to explain the generators and defining relations of groups. Due to the simplicity of its construction and high degree of symmetry, the Cayley graph has been paid significant attention from scholars in graph theory. In 1938, R. Frucht proved that for any given abstract group, there exists a graph in which that group is automorphism group. This important work initiated a new research field—groups and graphs, which focuses on the automorphism groups of graphs, the actions of groups on graphs, and graphs with various transitive properties, while also utilizing graphs to solve some group theory problems. Over the past 80 years, groups and graphs have developed into a significant area of algebraic graph theory, yielding many remarkable theories. In this paper, under the condition of odd order and twice prime valency, by means of the classification results of quasi-primitive graphs, through the analysis of the structure of the subgroups of the auto-morphism group of $\Gamma$ and its point stabilizer, and by using the classification results of primitive groups, we construct a class of locally primitive quasi-primitive graphs.