一类几乎单型的拟本原图的刻画

The Characterization of a Class of Quasiprimitive Graphs Admitting an Almost Simple Group

摘要: 设$\Gamma$ 是一个连通图，$G\le Aut\Gamma$ ，$\Gamma$ 是$G$ -边传递但不是$\left(G,2\right)$ -弧传递的。在奇数阶2倍素数度图的研究基础上，本文聚焦于拟本原非2-弧传递的情况，通过研究几乎单群$G$ 作用在$V$ 上的拟本原情形，对奇素数幂阶2倍素数度非2-弧传递图展开刻画。研究发现，此类图的结构较为特殊，要么是完全图$K_7$ 或$K_{11}$ ，要么同构于一个27阶10度图。这一结论进一步丰富了图论中关于特殊度数和传递性图的分类成果，为后续相关研究提供了重要参考。

Abstract: Let $\Gamma$ be a connected graph, $G\le Aut\Gamma$ , and $\Gamma$ be $G$ -edge-transitive but not $\left(G,2\right)$ -arc-transitive. Based on the research of graphs with odd order and twice prime valency, this paper focuses on the quasiprimitive non-2-arc-transitive case. By investigating the quasiprimitive action of almost simple groups $G$ on $V$ , we provide a characterization of odd prime power order graphs with twice prime valency that are non-2-arc-transitive. The research shows that the structures of such graphs are rather special. They are either complete graphs $K_7$ or $K_{11}$ , or isomorphic to a graph of order 27 and valency 10. This conclusion further enriches the classification results of graphs with special valency and transitivity in graph theory, and provides an important reference for subsequent related research.