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This paper is an endeavor to discuss some properties of zero-divisor graphsof the ring $\mathbb{Z}_n$, the ring of integers modulo $n$. The zero divisorgraph of a commutative ring $R$, is an undirected graph whose vertices are thenonzero zero-divisors of $R$, where two distinct vertices are adjacent if theirproduct is zero. The zero divisor graph of $R$ is denoted by $\Gamma(R)$. Wediscussed $\Gamma(\mathbb{Z}_n)$'s by the attributes of completeness, k-partitestructure, complete k-partite structure, regularity, chordality, $\gamma -\beta$ perfectness, simplicial vertices. The clique number for arbitrary$\Gamma(\mathbb{Z}_n)$ was also found. This work also explores relatedattributes of finite products$\Gamma(\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k})$, seeking to extendcertain results to the product rings. We find all$\Gamma(\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k})$ that are perfect.Likewise, a lower bound of clique number of$\Gamma(\mathbb{Z}_m\times\mathbb{Z}_n)$ was found. Later, in this paper wediscuss some properties of the zero divisor graph of the poset $D_n$, the setof positive divisors of a positive integer $n$ partially ordered bydivisibility.

Zero-Divisor Graphs of $\mathbb{Z}_n$, their products and $D_n$