Double-toroidal and 1-planar non-commuting graph of a group

Abstract

Let G be a finite non-abelian group and denote by Z(G) its center. The non-commuting graph of G on a transversal of the center is the graph whose vertices are the non-central elements of a transversal of Z(G) in G and two vertices x and y are adjacent whenever xy=yx. In this work, we classify the finite non-abelian groups whose non-commuting graph on a transversal of the centeris double-toroidal or 1-planar.