Extended total graph associated to finite commutative rings
UDC 512.5 For a commutative ring $R$  with nonzero identity $1\neq 0$, let $Z(R)$ denote the set of zero divisors. The total graph of $R$ denoted by $T_{\Gamma}(R)$ is a simple graph in which all elements of $R$ are vertices and any two distinct vertices $x$ an...
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| Date: | 2024 |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7494 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512676034117632 |
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| author | Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. |
| author_facet | Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. |
| author_sort | Altaf, Aaqib |
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| datestamp_date | 2024-07-15T03:05:04Z |
| description | UDC 512.5
For a commutative ring $R$  with nonzero identity $1\neq 0$, let $Z(R)$ denote the set of zero divisors. The total graph of $R$ denoted by $T_{\Gamma}(R)$ 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 $x+y\in Z(R)$. In this paper, we define an extension of the total graph denoted by $T(\Gamma^{e}(R))$ with vertex set as  $Z(R),$ and two distinct vertices $x$ and $y$ are adjacent if and only if  $x+y\in Z^*(R)$, where $ Z^{*}(R)$ is the set of nonzero zero divisors of $R$.  Our main aim is to characterize the finite commutative rings  whose $T(\Gamma^{e}(R))$ has clique numbers  $1,2,$ and $3$.  In addition, we characterize finite commutative nonlocal rings $R$ for which  the corresponding graph $T(\Gamma^{e}(R))$ has the clique number $4.$ |
| doi_str_mv | 10.3842/umzh.v76i5.7494 |
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| id | umjimathkievua-article-7494 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T03:32:34Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-74942024-07-15T03:05:04Z Extended total graph associated to finite commutative rings Extended total graph associated to finite commutative rings Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. Finite commutative rings Total graph Clique number UDC 512.5 For a commutative ring $R$  with nonzero identity $1\neq 0$, let $Z(R)$ denote the set of zero divisors. The total graph of $R$ denoted by $T_{\Gamma}(R)$ 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 $x+y\in Z(R)$. In this paper, we define an extension of the total graph denoted by $T(\Gamma^{e}(R))$ with vertex set as  $Z(R),$ and two distinct vertices $x$ and $y$ are adjacent if and only if  $x+y\in Z^*(R)$, where $ Z^{*}(R)$ is the set of nonzero zero divisors of $R$.  Our main aim is to characterize the finite commutative rings  whose $T(\Gamma^{e}(R))$ has clique numbers  $1,2,$ and $3$.  In addition, we characterize finite commutative nonlocal rings $R$ for which  the corresponding graph $T(\Gamma^{e}(R))$ has the clique number $4.$ УДК 512.5 Розширений тотальний граф, асоційований зі скінченними комутативними кільцями Нехай $Z(R)$ позначає множину дільників нуля для комутативного кільця $R$ з відмінною від нуля тотожністю $1\neq 0.$  Повний граф кільця $R$, який ми позначаємо $T_{\Gamma}(R)$, є простим графом, в якому всі елементи $R$ є вершинами, а дві різні вершини $x$ і $y$ суміжні  тоді і тільки тоді, коли $x+y\in Z(R)$.  У цій статті ми визначаємо розширення тотального графа, який ми позначаємо $T(\Gamma^{e}(R))$ і який має вершину $Z(R),$ а дві різні вершини $x$ і $ y$ суміжні тоді і тільки тоді, коли $x+y\in Z^*(R)$, де $Z^{*}(R)$ --- набір  ненульових дільників нуля в $R$.  Основною метою статті є характеристика скінченних комутативних кілець, у яких $T(\Gamma^{e}(R))$ має клікові числа $1,$ $2$ і $3$.  Крім того, охарактеризовано скінченні комутативні нелокальні кільця $R$, відповідний граф $T(\Gamma^{e}(R))$ яких має клікове число $4.$ Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7494 10.3842/umzh.v76i5.7494 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 791–801 Український математичний журнал; Том 76 № 6 (2024); 791–801 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7494/10026 Copyright (c) 2024 Shariefuddin Pirzada |
| spellingShingle | Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. Altaf, Aaqib Pirzada, S. Alghamdi, Ahmad M. Almotairi, Eman S. Extended total graph associated to finite commutative rings |
| title | Extended total graph associated to finite commutative rings |
| title_alt | Extended total graph associated to finite commutative rings |
| title_full | Extended total graph associated to finite commutative rings |
| title_fullStr | Extended total graph associated to finite commutative rings |
| title_full_unstemmed | Extended total graph associated to finite commutative rings |
| title_short | Extended total graph associated to finite commutative rings |
| title_sort | extended total graph associated to finite commutative rings |
| topic_facet | Finite commutative rings Total graph Clique number |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7494 |
| work_keys_str_mv | AT altafaaqib extendedtotalgraphassociatedtofinitecommutativerings AT pirzadas extendedtotalgraphassociatedtofinitecommutativerings AT alghamdiahmadm extendedtotalgraphassociatedtofinitecommutativerings AT almotairiemans extendedtotalgraphassociatedtofinitecommutativerings AT altafaaqib extendedtotalgraphassociatedtofinitecommutativerings AT pirzadas extendedtotalgraphassociatedtofinitecommutativerings AT alghamdiahmadm extendedtotalgraphassociatedtofinitecommutativerings AT almotairiemans extendedtotalgraphassociatedtofinitecommutativerings |