Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by T^n_R, with R^n-0 as the vertex set and two distinct vertices X and Y in R^n being adjacent if and only if there exists an nxn lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that X^T AY = 0 or Y ^TAX = 0, where, for a matrix B, B^T is the matrix transpose of B. If n = 1, then T^n_R is isomorphic to the zero divisor graph T(R), and so T^n_R is a generalization of T(R) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of T^n_R. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

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