The Line Graph Associated to the Total Graph of a Commutative Ring

Abstract

Let \(R\) be a commutative ring with identity and \(T(\Gamma(R))\) its total graph. The subject of this article is the investigation of the properties of the corresponding line graph \(L(T(\Gamma(R)))\). The classification of all commutative rings whose line graphs are planar or toroidal is given. It is shown that for every integer \(g \geq 0\) there are only finitely many commutative rings such that \(\gamma(L(T(\Gamma(R)))) = g\).