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

Abstract

This paper is an endeavor to discuss some properties of zero-divisor graphs of the ring $\mathbb{Z}_n$, the ring of integers modulo $n$. The zero divisor graph of a commutative ring $R$, is an undirected graph whose vertices are the nonzero zero-divisors of $R$, where two distinct vertices are adjacent if their product is zero. The zero divisor graph of $R$ is denoted by $\Gamma(R)$. We discussed $\Gamma(\mathbb{Z}_n)$'s by the attributes of completeness, k-partite structure, 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 related attributes of finite products $\Gamma(\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k})$, seeking to extend certain 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 we discuss some properties of the zero divisor graph of the poset $D_n$, the set of positive divisors of a positive integer $n$ partially ordered by divisibility.