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

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.