Summary: Let \(R\) be a commutative ring with the non-zero identity and \(n\) be a natural number. \(\Gamma ^n_R\) is a simple graph with \(R^n\setminus\{0\}\) as the vertex set and two distinct vertices \(X\) and \(Y\) in \(R^{n}\) are adjacent if and only if there exists an \(n \times n\) lower triangular matrix \(A\) over \(R\) whose entries on the main diagonal are non-zero such that \(AX^{t}=Y^{t}\) or \(AY^{t}=X^{t}\), where, for a matrix \(B\), \(B^{t}\) is the matrix transpose of \(B\). \(\Gamma^n_R\) is a generalization of Cayley graph. Let \(T_n (R)\) denote the \(n \times n\) upper triangular matrix ring over \(R\). In this paper, for an arbitrary ring \(R\), we investigate the properties of the graph \(\Gamma^n_{T_n(R)}\).