Aaqib Altaf, S. Pirzada, Ahmad M. Alghamdi, Eman S. Almotairi
{"title":"Extended total graph associated to finite commutative rings","authors":"Aaqib Altaf, S. Pirzada, Ahmad M. Alghamdi, Eman S. Almotairi","doi":"10.3842/umzh.v76i5.7494","DOIUrl":null,"url":null,"abstract":"<jats:p>UDC 512.5\nFor a commutative ring <mml:math>\n<mml:mrow>\n\t<mml:mi>R</mml:mi>\n</mml:mrow>\n</mml:math> with nonzero identity <mml:math>\n<mml:mrow>\n\t<mml:mn>1</mml:mn>\n\t<mml:mo>≠</mml:mo>\n\t<mml:mn>0</mml:mn>\n</mml:mrow>\n</mml:math>, let <mml:math>\n<mml:mrow>\n\t<mml:mi>Z</mml:mi>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math> denote the set of zero divisors. The total graph of <mml:math>\n<mml:mrow>\n\t<mml:mi>R</mml:mi>\n</mml:mrow>\n</mml:math> denoted by <mml:math>\n<mml:mrow>\n\t<mml:msub>\n\t\t<mml:mi>T</mml:mi>\n\t\t<mml:mi>Γ</mml:mi>\n\t</mml:msub>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math> is a simple graph in which all elements of <mml:math>\n<mml:mrow>\n\t<mml:mi>R</mml:mi>\n</mml:mrow>\n</mml:math> are vertices and any two distinct vertices <mml:math>\n<mml:mrow>\n\t<mml:mi>x</mml:mi>\n</mml:mrow>\n</mml:math> and <mml:math>\n<mml:mrow>\n\t<mml:mi>y</mml:mi>\n</mml:mrow>\n</mml:math> are adjacent if and only if <mml:math>\n<mml:mrow>\n\t<mml:mi>x</mml:mi>\n\t<mml:mo>+</mml:mo>\n\t<mml:mi>y</mml:mi>\n\t<mml:mo>∈</mml:mo>\n\t<mml:mi>Z</mml:mi>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math>. In this paper, we define an extension of the total graph denoted by <mml:math>\n<mml:mrow>\n\t<mml:mi>T</mml:mi>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:msup>\n\t\t\t<mml:mi>Γ</mml:mi>\n\t\t\t<mml:mi>e</mml:mi>\n\t\t</mml:msup>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math> with vertex set as <mml:math>\n<mml:mrow>\n\t<mml:mi>Z</mml:mi>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n\t<mml:mo>,</mml:mo>\n</mml:mrow>\n</mml:math> and two distinct vertices <mml:math>\n<mml:mrow>\n\t<mml:mi>x</mml:mi>\n</mml:mrow>\n</mml:math> and <mml:math>\n<mml:mrow>\n\t<mml:mi>y</mml:mi>\n</mml:mrow>\n</mml:math> are adjacent if and only if <mml:math>\n<mml:mrow>\n\t<mml:mi>x</mml:mi>\n\t<mml:mo>+</mml:mo>\n\t<mml:mi>y</mml:mi>\n\t<mml:mo>∈</mml:mo>\n\t<mml:msup>\n\t\t<mml:mi>Z</mml:mi>\n\t\t<mml:mo>*</mml:mo>\n\t</mml:msup>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math>, where <mml:math>\n<mml:mrow>\n\t<mml:msup>\n\t\t<mml:mi>Z</mml:mi>\n\t\t<mml:mo>*</mml:mo>\n\t</mml:msup>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math> is the set of nonzero zero divisors of <mml:math>\n<mml:mrow>\n\t<mml:mi>R</mml:mi>\n</mml:mrow>\n</mml:math>. Our main aim is to characterize the finite commutative rings whose <mml:math>\n<mml:mrow>\n\t<mml:mi>T</mml:mi>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:msup>\n\t\t\t<mml:mi>Γ</mml:mi>\n\t\t\t<mml:mi>e</mml:mi>\n\t\t</mml:msup>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math> has clique numbers <mml:math>\n<mml:mrow>\n\t<mml:mn>1,2</mml:mn>\n\t<mml:mo>,</mml:mo>\n</mml:mrow>\n</mml:math> and <mml:math>\n<mml:mrow>\n\t<mml:mn>3</mml:mn>\n</mml:mrow>\n</mml:math>. In addition, we characterize finite commutative nonlocal rings <mml:math>\n<mml:mrow>\n\t<mml:mi>R</mml:mi>\n</mml:mrow>\n</mml:math> for which the corresponding graph <mml:math>\n<mml:mrow>\n\t<mml:mi>T</mml:mi>\n\t<mml:mrow>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:msup>\n\t\t\t<mml:mi>Γ</mml:mi>\n\t\t\t<mml:mi>e</mml:mi>\n\t\t</mml:msup>\n\t\t<mml:mo form=\"prefix\">(</mml:mo>\n\t\t<mml:mi>R</mml:mi>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t\t<mml:mo form=\"postfix\">)</mml:mo>\n\t</mml:mrow>\n</mml:mrow>\n</mml:math> has the clique number <mml:math>\n<mml:mrow>\n\t<mml:mn>4.</mml:mn>\n</mml:mrow>\n</mml:math></jats:p>","PeriodicalId":163365,"journal":{"name":"Ukrains’kyi Matematychnyi Zhurnal","volume":" 28","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ukrains’kyi Matematychnyi Zhurnal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3842/umzh.v76i5.7494","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
UDC 512.5
For a commutative ring R with nonzero identity 1≠0, let Z(R) denote the set of zero divisors. The total graph of R denoted by TΓ(R) is a simple graph in which all elements of R are vertices and any two distinct vertices x and y are adjacent if and only if x+y∈Z(R). In this paper, we define an extension of the total graph denoted by T(Γe(R)) with vertex set as Z(R), and two distinct vertices x and y are adjacent if and only if x+y∈Z*(R), where Z*(R) is the set of nonzero zero divisors of R. Our main aim is to characterize the finite commutative rings whose T(Γe(R)) has clique numbers 1,2, and 3. In addition, we characterize finite commutative nonlocal rings R for which the corresponding graph T(Γe(R)) has the clique number 4.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
