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 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∖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.