A completeness criterion for the common divisor graph on $p$-regular class sizes

Abstract: Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $\Gamma_p(G)$ 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 $\Gamma_p(G)$ is a $k$-regular graph with $k\geq 1$, then it is a complete graph with $k+1$ vertices. We also pose a conjecture regarding the order of products of $p$-regular elements with coprime conjugacy class sizes, whose validity would enable to drop the $p$-separability hypothesis.