M AFKHAMI
Articles written in Proceedings – Mathematical Sciences
Volume 128 Issue 1 February 2018 Article ID 0009 Research Article
A generalization of zero divisor graphs associated to commutative rings
M AFKHAMI A ERFANIAN K KHASHYARMANESH N VAEZ MOOSAVI
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 $X^{T} AY = 0$ or $Y^{T} AX = 0$, 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.
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