The commuting graphs of certain cyclic-by-abelian groups

Abstract: Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or the disjoint union of $|G'| + 1$ complete graphs. These results apply to all finite metacyclic groups, and to groups of square-free order in particular.