A Note on the Double Total Graph \(T_u(\Gamma(R))\) and \(T_u(\Gamma(\mathbb{Z}_n \times \mathbb{Z}_m))\)

Abstract

Considering a commutative ring \(R\) with unity as the set of vertices and two vertices \(x\) and \(y\) are adjacent if and only if \(u+(x+y) \in Z(R)\) for some \(u \in U(R)\), the resulting graph \(T_{u}(\Gamma(R))\) is known as the double total graph. In this paper we find the degree of any vertex in \(T_{u}(\Gamma(R))\) for a weakly unit fusible ring \(R\) and domination number of \(T_{u}(\Gamma(R))\) for any ring \(R\). Also, we investigate the properties of \(T_{u}(\Gamma(\mathbb{Z}_{n}\times\mathbb{Z}_{m}))\) and characterize $R$ in terms of toroidal \(T_{u}(\Gamma(R))\).