论文标题
$ \ mathbb {z} _n $的零划分图，他们的产品和$ d_n $
Zero-Divisor Graphs of $\mathbb{Z}_n$, their products and $D_n$
论文作者
论文摘要
本文是讨论环$ \ mathbb {z} _n $的零分量图的某些属性的努力。通勤环$ r $的零除数图是一个无向图的图形，其顶点是$ r $的非零零分数，如果它们的产品为零，则两个不同的顶点相邻。 $ r $的零除数图用$γ（r）$表示。我们通过完整性，K-党结构，完整的K-Partite结构，规律性，弦，$γ-β$ Perfectness，Simplixical Pertices的属性讨论了$γ（\ mathbb {z} _n）$。还找到了任意$γ（\ mathbb {z} _n）$的集团编号。这项工作还探讨了有限产物的相关属性$γ（\ Mathbb {z} _ {n_1} \ times \ cdots \ cdots \ times \ times \ times \ mathbb {z} _ {n_k}）$，旨在将某些结果扩展到产品环上。我们找到所有$γ（\ Mathbb {z} _ {n_1} \ times \ cdots \ times \ times \ times \ mathbb {z} _ {n_k}）$是完美的。同样，找到了$γ（\ mathbb {z} _m \ times \ times \ mathbb {z} _n）$的集团数量的下限。后来，在本文中，我们讨论了poset $ d_n $的零除数图的某些属性，$ d_n $，这是一个正整数$ n $的正面除数部分，该$ n $部分按分裂性订购。
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 $Γ(R)$. We discussed $Γ(\mathbb{Z}_n)$'s by the attributes of completeness, k-partite structure, complete k-partite structure, regularity, chordality, $γ- β$ perfectness, simplicial vertices. The clique number for arbitrary $Γ(\mathbb{Z}_n)$ was also found. This work also explores related attributes of finite products $Γ(\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k})$, seeking to extend certain results to the product rings. We find all $Γ(\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k})$ that are perfect. Likewise, a lower bound of clique number of $Γ(\mathbb{Z}_m\times\mathbb{Z}_n)$ 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.
