一类4度无平方因子阶边传递图的分类

Classification of Tetravalent Edge-Transitive Graphs of Square-Free Order

摘要: 设$\Gamma$ 是一个无平方因子阶的4度连通图，且$G\le Aut\Gamma$ 是几乎单群，$G$ 在顶点集上作用是本原的且在边集上是传递的，若图是非2-弧传递的，则存在唯一性。本文研究了此类图的结构与分类问题，通过引入陪集图构造，结合本原置换群理论及已知的有限单群分类结果，我们完整刻画了所有满足条件的图。本文的结果推广了无平方因子阶对称图的相关研究，并为更高度数图的分类提供了参考。

Abstract: Let $\Gamma$ be a finite connected tetravalent graph of square-free order, and let $G\le Aut\Gamma$ be an almost simple group acting primitively on the vertex set and transitively on the edge set. If the graph is not $\left(G,2\right)$ -arc-transitive, then it is unique. This paper investigates the structure and classification of such graphs. By introducing coset graph constructions and combining the theory of primitive permutation groups with known classifications of finite simple groups, we completely characterize all graphs satisfying these conditions. The results of this paper generalize related studies on symmetric graphs of square-free order and provide a reference for the classification of graphs of higher valency.