The Zero-divisor Graphs of Posets and an Application to Semigroups
The Zero-divisor Graphs of Posets and an Application to Semigroups
复制标题
DOI：
10.1007/s00373-010-0955-4
复制
发表时间：
2010-11
影响因子：
0.7
通讯作者：
D. Lu;Tongsuo Wu
中科院分区：
文献类型：
--
作者：
D. Lu;Tongsuo Wu
在本文中，我们引入了紧图的概念。我们证明了一个简单图是一个紧图当且仅当G是一个偏序集的零因子图，并给出了Halaeland和Jukl（离散数学309：4584-4589，2009）的主要结果的一个新的证明，即如果G是一个偏序集的零因子图，那么在一个温和的假设下，G的色数和团数重合。我们观察到约化交换半群（环）的零因子图是紧的，从而提供了一大类可以实现为偏序集的零因子图的图G。此外，利用这些结果，我们分别给出了偏序集的零因子图和含0的既约交换半群的零因子图的等价刻画。
In this paper, we introduce the notion of a compact graph. We show that a simple graph is a compact graph if and only ifGis the zero-divisor graph of a poset, and give a new proof of the main result in Halaš and Jukl (Discrete Math 309:4584–4589, 2009) stating that ifGis the zero-divisor graph of a poset, then the chromatic number and the clique number ofGcoincide under a mild assumption. We observe that the zero-divisor graphs of reduced commutative semigroups (rings) are compact, thus provide a large class of graphsGthat could be realized as zero-divisor graphs of posets. In addition, using these results, we give some equivalent descriptions for the zero-divisor graphs of posets and reduced commutative semigroups with 0 respectively.