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

TL;DR

This paper investigates the properties of zero-divisor graphs of rings of integers modulo n, their products, and the divisor poset, providing new insights into their structure, clique numbers, and perfectness.

Contribution

It introduces new characterizations of zero-divisor graphs for $\

Findings

Determined clique numbers for various zero-divisor graphs.

Identified conditions for perfectness of product graphs.

Analyzed structural properties like completeness and chordality.

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.