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

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.