  Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$
on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set
with $x,y\in X$ joined by an edge whenever $x\neq y$ and $xy=yx$. Let $T$ be a
transversal of the center $Z(G)$ of $G$. When $G$ is a finite non-abelian group
and $X=T\setminus Z(G)$, we denote the graph ${\cal C}(G,X)$ by ${\cal T}(G)$.
In this paper, we show that ${\cal T}(G)$ is a connected strongly regular graph
if and only if $G$ is isoclinic to an extraspecial $2$-group of order at least
$32$. We also characterize the finite non-abelian groups $G$ for which the
graph ${\cal T}(G)$ is disconnected strongly regular.
