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Morgan and Parker have proved that if $G$ is a group satisfying the conditionthat $Z(G) = 1$, then the connected components of the commuting graph of $G$have diameter at most $10$. Parker has proved that if in addition $G$ issolvable, then the commuting graph of $G$ is disconnected if and only if $G$ isa 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 alsoprove 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.

Extending Morgan and Parker's results about commuting graphs