Groups whose non-commuting graph on a transversal is planar or toroidal

Abstract

Let [Formula: see text] be a finite non-abelian group and let [Formula: see text] be a transversal of the center of [Formula: see text]. The non-commuting graph of [Formula: see text] on a transversal of the center is the graph denoted by [Formula: see text] whose vertices are the non-central elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are joined by an edge whenever [Formula: see text]. In this paper, we determine (up to isoclinism) all finite non-abelian groups whose non-commuting graph on a transversal of the center is planar or toroidal.