Finite groups whose commuting graph is split

Abstract

As a contribution to the study of graphs defined on groups, we show that for a finite group G the following statements are equivalent: the commuting graph of G is a split graph; the commuting graph of G is a threshold graph; either G is abelian, or G is a generalized dihedral group D(A)=⟨A,t:(∀a∈A)(at)2=1⟩ where A is an abelian group of odd order.