Let R be a commutative ring with the non-zero identity and n be a natural number. T^n R is a simple graph with R^n \ {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 × 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 = Xt, where, for a matrix B, B^t is the matrix transpose of B. T^n R is a generalization of Cayley graph. Let T^n(R) denote the n × n upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph T^n_{T^n(R)}.

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