{"title":"On the zero-divisor hypergraph of a reduced ring","authors":"T. Asir, A. Kumar, A. Mehdi","doi":"10.1007/s10474-023-01362-2","DOIUrl":null,"url":null,"abstract":"<div><p>The concept of zero-divisor graphs of rings is widely used for establishing relationships between the properties of graphs and the properties of the underlying ring. The zero-divisor graph of a ring is generalized to the <i>k</i>-zero-divisor hypergraph of a ring <i>R</i> for <span>\\(k\\in \\mathbb{N}\\)</span>, which is denoted by <span>\\(\\mathcal{H}_{k}(R)\\)</span>.\nThis paper is an endeavor to discuss some properties of zero-divisor hypergraphs.\nWe determine the diameter and girth of <span>\\(\\mathcal{H}_{k}(R)\\)</span> whenever <i>R</i> is reduced.\nAlso, we characterize all commutative rings <i>R</i> for which <span>\\(\\mathcal{H}_{k}(R)\\)</span> is in some known class of graphs.\nFurther, we obtain certain necessary conditions for <span>\\(\\mathcal{H}_{k}(R)\\)</span> to be a Hamilton Berge cycle and a flag-traversing tour.\nMoreover, we answer a case of the question raised by Eslahchi et al. [15].</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10474-023-01362-2.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-023-01362-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The concept of zero-divisor graphs of rings is widely used for establishing relationships between the properties of graphs and the properties of the underlying ring. The zero-divisor graph of a ring is generalized to the k-zero-divisor hypergraph of a ring R for \(k\in \mathbb{N}\), which is denoted by \(\mathcal{H}_{k}(R)\).
This paper is an endeavor to discuss some properties of zero-divisor hypergraphs.
We determine the diameter and girth of \(\mathcal{H}_{k}(R)\) whenever R is reduced.
Also, we characterize all commutative rings R for which \(\mathcal{H}_{k}(R)\) is in some known class of graphs.
Further, we obtain certain necessary conditions for \(\mathcal{H}_{k}(R)\) to be a Hamilton Berge cycle and a flag-traversing tour.
Moreover, we answer a case of the question raised by Eslahchi et al. [15].
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.