Conjugacy classes of finite groups and graph regularity

Given a finite group $G$, denote by $Γ(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $Γ(G)$ to be adjacent if and only if they are not coprime numbers. In this note we prove that, if $Γ(G)$ is a $k$-regular graph with $k\geq 1$, then $Γ(G)$ is a complete graph with $k+1$ vertices.

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