Conjugacy classes of finite groups and graph regularity

Abstract

Abstract Given a finite group G , denote by

Γ ( G )

$\Gamma (G)$

the simple

undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of G , and set two vertices of Γ ( G )

$\Gamma (G)$

to be

adjacent if and only if they are not coprime numbers. In this note we prove that, if Γ ( G )

$\Gamma (G)$

is a k -regular graph with k ≥ 1, then

Γ ( G )

$\Gamma (G)$

is a complete graph with

k + 1

$k+1$

vertices.