A Note on the Double Total Graph \(T_u(Γ(R))\) and \(T_u(Γ(\mathbb{Z}_n × \mathbb{Z}_m))\) - Singh, Ngangom Rojitkumar; Dutta, Sanghita (Communications in Mathematics and Applications, 2021)
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 \(T_{u}(Γ(R))\) is known as the double total graph. In this paper we find the degree of any vertex in \(T_{u}(Γ(R))\) for a weakly unit fusible ring \(R\) and domination number of \(T_{u}(Γ(R))\) for any ring \(R\). Also, we investigate the properties of \(T_{u}(Γ(\mathbb{Z}_{n}×\mathbb{Z}_{m}))\) and characterize $R$ in terms of toroidal \(T_{u}(Γ(R))\).
Keywords: Fusible ring; Weakly unit fusible ring; Unit graph; Total graph; Double total graph