The commuting graphs of certain cyclic-by-abelian groups

Let G be a finite, non-abelian group of the form $G = A N$, where A ≤ G is abelian, and N G is cyclic. We prove that the commuting graph Γ(G) of G is either a connected graph of diameter at most four, or the disjoint union of several complete graphs. These results apply to all finite metacyclic groups, and groups of square-free order in particular.