奇数阶2倍素数度的局部本原图的刻画

Characterization of Locally Primitive Graphs of Odd Order with Twice Prime Valency

摘要: 局部本原图是图论与群论交叉研究的重要对象，在图的结构和对称性研究中意义重大。本文在已有研究基础上，针对奇数阶2倍素数度的局部本原但非拟本原图展开深入研究。研究证明，此类图是某一商图的正规覆盖。并进一步研究了其自同构群关于极大非传递正规子群的商群，要么是几乎单的，要么其基柱同构于$Z_p^k$ ，其中$p$ 是素数，$k\le r$ 。

Abstract: Locally primitive graphs are important objects in the intersection of graph theory and group theory, and they are of great significance in the study of the structure and symmetry of graphs. Based on previous research, this paper conducts an in-depth study on locally primitive but non-quasiprimitive graphs with odd order and twice prime valency. The research proves that such graphs are normal covers of a certain quotient graphs. Furthermore, it is investigated that the quotient group of the automorphism group of these graphs with respect to a maximal non-transitive normal subgroup is either almost simple or has a socle isomorphic to $Z_p^k$ , where p is a prime number, $k\le r$ .