A generalization of total graphs

Let R be a commutative ring with nonzero identity, (L_{n}(R)) be the set of all lower triangular (ntimes n) 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},dots ,x_{n})) and ((y_{1},dots ,y_{n})) are adjacent if and only if ((x_{1}+y_{1}, ldots ,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}{setminus } U)) and (GT^{n}_{U}(R^n)).