  Let $G$ be a finite group. Recall that an $A$-group is a group whose Sylow
subgroups are all abelian. In this paper, we investigate the upper bound on the
diameter of the commuting graph of a solvable $A$-group. Assuming that the
commuting graph is connected, we show when the derived length of $G$ is 2, the
diameter of the commuting graph will be at most 4. In the general case, we show
that the diameter of the commuting graph will be at most 6. In both cases,
examples are provided to show that the upper bound of the commuting graph
cannot be improved.
