6度边本原双本原图的分类

Classification of 6-Valent Edge-Primitive and Bi-Primitive Graphs

摘要: 如果$\Gamma =\left(V,E\right)$ 是一个6度连通的二部图，其中$V=U\cup W$ ，令$G\le Aut\Gamma$ ，若$G$ 中存在一个指数为2的正规子群$G^{+}$ ，且$G^{+}$ 在两个分部$U$ 和$W$ 上作用是本原的，则我们称图$\Gamma$ 是$G$ -双本原的。本文通过对6度边本原图的分类，确定了在非忠实的作用下，6度图中存在的双本原图只有$K_{6,6}$ ，并进一步了解6度双本原图的结构和性质。

Abstract: If $\Gamma =\left(V,E\right)$ is a 6-valent connected bipartite graph, where $V=U\cup W$ and $G\le Aut\Gamma$ , if here exists a normal subgroup $G^{+}$ of index 2 in $G$ , such that $G^{+}$ acts primitively on both parts $U$ and $W$ , then we say that the graph $\Gamma$ is $G$ -biprimitive. By classifying 6-valent edge-primitive graphs, this paper identifies that under non-faithful actions, the only biprimitive graph among 6-valent graphs is $K_{6,6}$ . Furthermore, the structure and properties of 6-valent bi-primitive graphs are further explored.