Let R be a commutative ring with nonzero identity, L_n(R) be the set of all lower triangular nxn matrices, and U be a triangular subset of R^n i.e. the product of any lower triangular matrix with the transpose of any element of U, belongs to U. The graph GT^n_U (R^n) is a simple graph whose vertices consists of all elements of R^n, and two distinct vertices (x_1, ... ,x_n) and (y_1,...,y_n) are adjacent if and only if (x_1 + y_1,..., x_n + y_n) in U. The graph GT^n_U (R^n) is a generalization for total graphs. In this paper, we investigate the basic properties of GT^n_U (R^n). Moreover, we study the planarity of the graphs GT^n_U (U), GT^n_U (R^n-U) and GT^n_U (R^n).

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