关于 $p$-正则类大小上公共除数图的完备性准则
A completeness criterion for the common divisor graph on $p$-regular class sizes
AI总结：
本文提出一个关于 $p$-正则类大小上公共除数图的完备性准则，证明在特定条件下该图必为完全图。
摘要：
设 $G$ 为有限群。对于某个固定的素数 $p$，令 $Γ_p(G)$ 是以 $G$ 中 $p$-正则共轭类大小集合为基础构建的公共除数图：这是一个简单无向图，其顶点是那些非中心元的类大小，且 $p$ 不整除其阶数。两个不同的顶点相邻当且仅当它们不互质。在本笔记中，我们证明如果 $Γ_p(G)$ 是一个 $k$-正则图，其中 $k≥1$，则它是一个 $k+1$ 个顶点的完全图。
英文摘要：
Let $G$ be a finite group. For some fixed prime $p$, let $Γ_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of those non-central elements of $G$ such that $p$ does not divide their order, and two distinct vertices are adjacent if and only if they are not coprime. In this note we prove that if $Γ_p(G)$ is a $k$-regular graph with $k\geq 1$, then it is a complete graph with $k+1$ vertices.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
