# On the planarity, genus, and crosscap of the weakly zero-divisor graph of commutative rings

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https://doi.org/10.33044/revuma.2837## Abstract

Let $R$ be a commutative ring and $Z(R)$ its zero-divisors set. The weakly zero-divisor graph of $R$, denoted by $W\Gamma(R)$, is an undirected graph with the nonzero zero-divisors $Z(R)^*$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exist $a \in \mathrm{Ann}(x)$ and $b \in \mathrm{Ann}(y)$ such that $ab = 0$. In this paper, we characterize finite rings $R$ for which the weakly zero-divisor graph $W\Gamma(R)$ belongs to some well-known families of graphs. Further, we classify the finite rings $R$ for which $W\Gamma(R)$ is planar, toroidal or double toroidal. Finally, we classify the finite rings $R$ for which the graph $W\Gamma(R)$ has crosscap at most two.

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Copyright (c) 2024 Nadeem ur Rehman, Mohd Nazim, Shabir Ahmad Mir

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