Abstract
Morgan and Parker proved that if G is a group with
${\textbf{Z}(G)} = 1$
, then the connected components of the commuting graph of G have diameter at most
$10$
. Parker 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
$G' \cap {\textbf{Z}(G)} = 1$
. We also prove that if G is solvable and
$G/{\textbf{Z}(G)}$
is either a Frobenius group or a
$2$
-Frobenius group, then the commuting graph of G is disconnected.
Non-commuting graph of the dihedral group determined by Hosoya parameters
