On groups whose commuting graph on a transversal is strongly regular

Abstract Given a finite group G and a subset X of G , the commuting graph of G on X , denoted by C ( G , X ) , is the graph that has X as its vertex set and two vertices x and y are joined by an edge whenever x ≠ y and x y = y x . Let T be a transversal of the center Z ( G ) of G . When G is a finite non-abelian group and X = T ∖ Z ( G ) , we denote the graph C ( G , X ) by T ( G ) . In this paper, we show that 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 T ( G ) is disconnected strongly regular.