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

如果 Γ=( V,E ) 是一个6度连通的二部图，其中 V=U∪W ，令 G≤AutΓ ，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 Γ 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。If Γ=( V,E ) is a 6-valent connected bipartite graph, where V=U∪W and G≤AutΓ , 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 Γ 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.