A generalization of zero divisor graphs associated to commutative rings

Abstract

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by $$\Gamma ^n_R$$ , with $$R^n\backslash \{0\}$$ as the vertex set and two distinct vertices X and Y in $$R^n$$ being adjacent if and only if there exists an $$n\times n$$ 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 $$X^TAY=0$$ or $$Y^TAX=0$$ , where, for a matrix $$B, B^T$$ is the matrix transpose of B. If $$n=1$$ , then $$\Gamma ^n_R$$ is isomorphic to the zero divisor graph $$\Gamma (R)$$ , and so $$\Gamma ^n_R$$ is a generalization of $$\Gamma (R)$$ which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of $$\Gamma ^n_ R$$ . We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.