交换环点积图补的一些结果

IF 0.5 Q3 MATHEMATICS

Asian-European Journal of Mathematics Pub Date : 2023-10-20 DOI:10.1142/s1793557123502157

S. Visweswaran

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引用次数: 0

摘要

设[公式:见文]是一个具有恒等的非零交换环。设[公式:见文本]为正整数。设[公式:见文]([公式:见文]次)。用[公式:见文]表示的[公式:见文]的总点积图是顶点集等于[公式:见文]的无向图，且不同顶点[公式:见文]与[公式:见文]相邻当且仅当[公式:见文]，其中[公式:见文]是[公式:见文]与[公式:见文]的法向点积。设[公式:见文]表示[公式:见文]的所有零因子的集合，并用[公式:见文]表示[公式:见文]。[公式:见文]的零因子点积图，用[公式:见文]表示，是由[公式:见文]引出的[公式:见文]的子图。图表[公式:见文](分别为[公式:见文])是由Badawi [common]引入并研究的。代数43(1)(2015)43 - 50 [j]。在本文中，我们对[公式:见文]进行表征，使[公式:见文](分别为[公式:见文])连通，并确定[公式:见文](分别为[公式:见文])在连通时的直径和半径。此外，如果[公式:见文](分别为[公式:见文])是连通的，那么我们对[公式:见文]进行表征，使[公式:见文](分别为[公式:见文])允许一个切顶点。

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Some results on the complement of the dot product graph of a commutative ring

Let [Formula: see text] be a nonzero commutative ring with identity. Let [Formula: see text] be a positive integer. Let [Formula: see text] ([Formula: see text] times). The total dot product graph of [Formula: see text] denoted by [Formula: see text] is an undirected graph with vertex set which equals [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text], where [Formula: see text] is the normal dot product of [Formula: see text] and [Formula: see text]. Let [Formula: see text] denote the set of all zero-divisors of [Formula: see text] and let us denote [Formula: see text] by [Formula: see text]. The zero-divisor dot product graph of [Formula: see text], denoted by [Formula: see text] is the subgraph of [Formula: see text] induced by [Formula: see text]. The graph [Formula: see text] (respectively, [Formula: see text]) was introduced and investigated by Badawi [Commun. Algebra 43(1) (2015) 43–50]. In this paper, we characterize [Formula: see text] such that [Formula: see text] (respectively, [Formula: see text]) is connected and determine the diameter and the radius of [Formula: see text] (respectively, [Formula: see text]) whenever it is connected. Moreover, if [Formula: see text] (respectively, [Formula: see text]) is connected, then we characterize [Formula: see text] such that [Formula: see text] (respectively, [Formula: see text]) admits a cut vertex.

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来源期刊

Asian-European Journal of Mathematics MATHEMATICS-

CiteScore

1.30

自引率

12.50%

发文量

169

期刊介绍： Asian-European Journal of Mathematics is an international journal which is devoted to original research in the field of pure and applied mathematics. The aim of the journal is to provide a medium by which new ideas can be discussed among researchers from diverse fields in mathematics. It publishes high quality research papers in the fields of contemporary pure and applied mathematics with a broad range of topics including algebra, analysis, topology, geometry, functional analysis, number theory, differential equations, operational research, combinatorics, theoretical statistics and probability, theoretical computer science and logic. Although the journal focuses on the original research articles, it also welcomes survey articles and short notes. All papers will be peer-reviewed within approximately four months.