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 \(xy\neq yx\). In this work, we classify the finite non-abelian groups whose non-commuting graph on a transversal of the center is double-toroidal or \(1\)-planar.