On the regularity of a graph related to conjugacy classes of groups

Abstract

Given a finite group G, denote by Γ. (G) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G, and for which two vertices of Γ. (G) are adjacent if and only if they are not coprime numbers. In this note we prove that Γ. (G) is a 2-regular graph if and only if it is a complete graph with three vertices, and Γ. (G) is a 3-regular graph if and only if it is a complete graph with four vertices. © 2012 Elsevier Ltd.