{"title":"Independent domination polynomial for the cozero divisor graph of the ring of integers modulo n","authors":"B. Rather","doi":"10.47443/dml.2023.215","DOIUrl":null,"url":null,"abstract":"The cozero divisor graph Γ (cid:48) ( R ) of a commutative ring R is a simple graph whose vertex set is the set of non-zero non-unit elements of R such that two distinct vertices x and y of Γ (cid:48) ( R ) are adjacent if and only if x / ∈ Ry and y / ∈ Rx , where Rx is the ideal generated by x . In this article, the independent domination polynomial of Γ (cid:48) ( Z n ) is found for n ∈ { p 1 p 2 , p 1 p 2 p 3 , p n 1 1 p 2 } , where p i ’s are primes, n 1 is an integer greater than 1 , and Z n is the integer modulo ring. It is shown that the independent domination polynomial of Γ (cid:48) ( Z p 1 p 2 ) has only one real root. It is also proved that these polynomials are not unimodal but are log-concave under certain conditions.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":"13 28","pages":""},"PeriodicalIF":16.4000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.47443/dml.2023.215","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The cozero divisor graph Γ (cid:48) ( R ) of a commutative ring R is a simple graph whose vertex set is the set of non-zero non-unit elements of R such that two distinct vertices x and y of Γ (cid:48) ( R ) are adjacent if and only if x / ∈ Ry and y / ∈ Rx , where Rx is the ideal generated by x . In this article, the independent domination polynomial of Γ (cid:48) ( Z n ) is found for n ∈ { p 1 p 2 , p 1 p 2 p 3 , p n 1 1 p 2 } , where p i ’s are primes, n 1 is an integer greater than 1 , and Z n is the integer modulo ring. It is shown that the independent domination polynomial of Γ (cid:48) ( Z p 1 p 2 ) has only one real root. It is also proved that these polynomials are not unimodal but are log-concave under certain conditions.
交换环 R 的共零除数图 Γ (cid:48) ( R ) 是一个简单图,其顶点集是 R 的非零非单位元素集,当且仅当 x /∈Ry 和 y /∈Rx 时,Γ (cid:48) ( R ) 的两个不同顶点 x 和 y 相邻,其中 Rx 是由 x 生成的理想。本文将为 n∈ { p 1 p 2 , p 1 p 2 p 3 , p n 1 1 p 2 } 求出 Γ (cid:48) ( Z n ) 的独立支配多项式。其中 p i 是素数,n 1 是大于 1 的整数,Z n 是整数模环。证明了Γ (cid:48) ( Z p 1 p 2 ) 的独立支配多项式只有一个实数根。还证明了这些多项式不是单模态的,但在某些条件下是对数凹的。
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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