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) =< A, t : ( for all a is an element of A)(at)(2 ) = 1 > where A is an abelian group of odd order.