# A generalization of zero divisor graphs associated to commutative rings

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## 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.

## Keywords

Zero divisor graph lower triangular matrix genus complete graph## 2010 Mathematics Subject Classification

15B33 05C10 05C25 05C45## References

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