Finite groups whose commuting graph is split

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 $D(A)=\langle A,t:(\forall a\in A)(at)^2=1\rangle$ where $A$ is an abelian group of odd order.