某些交换环的单位积图
Unity Product Graph of Some Commutative Rings
摘要：
图是一种工具，它被广泛用于模拟不同领域的各种问题。到目前为止，已经开发了许多图来表示代数结构，特别是环，以研究它们的性质。本文以交换环$ R $为研究对象，引入了与$ R $及其补环相关的单位积图的概念.此外，我们证明了如果单位积图的顶点数至少为2，则该图是不连通的，而它的补图是连通的。进一步证明了存在某些交换环，如布尔环和布尔环的笛卡儿积，其中它们的单位积图是平凡的。从而建立了确定单位积图孤立点个数的一些结果。我们还刻画了单位的交换环，其中它们的单位积图和补单位积图分别是空图和完全图。最后，我们证明了单位积图及其补图的围长、直径、半径、控制数、色数、团数以及平面性和Hamilton性等性质。
英文摘要：
A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In this article, by focusing on commutative ring $ R $, we introduce a new notion of unity product graph associated with $ R $ and its complement. In addition, we prove that if the number of vertices of the unity product graph is at least 2, then the graph is disconnected, while its complement graph is connected. Furthermore, it is shown that there are some commutative rings with such as Boolean ring and the Cartesian product of Boolean rings in which their associated unity product graphs are trivial. Consequently, some results are established to determine the number of isolated vertices in unity product graph. We also characterize commutative rings with unity in which their associated unity product and complement unity product graphs are empty graph and complete graph, respectively. Finally, we prove some results on the properties of the unity product graph and its complement in terms of girth, diameter, radius, dominating number, chromatic number and clique number as well as planarity and Hamiltonian.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
