  Morgan and Parker have proved that if $G$ is a group satisfying the condition
that $Z(G) = 1$, then the connected components of the commuting graph of $G$
have diameter at most $10$. Parker has proved that if in addition $G$ is
solvable, then the commuting graph of $G$ is disconnected if and only if $G$ is
a Frobenius group or a $2$-Frobenius group, and if the commuting graph of $G$
is connected, then its diameter is at most $8$. We prove that the hypothesis $Z
(G) = 1$ in these results can be replaced with $G' \cap Z(G) = 1$. We also
prove that if $G$ is solvable and $G/Z(G)$ is either a Frobenius group or a
$2$-Frobenius group, then the commuting graph of $G$ is disconnected.
