An extension of the essential graph of a ring
Let \(A\) be a commutative ring with non-zero identity, and \(E(A)=\{p\in A | ann_A(pq)\leq_e A, ~\mbox { for some } ~q\in A^* \}\) . The extended essential graph, denoted by \(E_gG(A)\) is a graph with the vertex set \(E(A)^*=E(A)\setminus\{0\}\). Two distinct vertices \(r, s\in E(A)^*\) are adjace...
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| Дата: | 2024 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2024
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2120 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-21202024-04-21T17:47:57Z An extension of the essential graph of a ring Ali, Asma Ahmad, Bakhtiyar zero-divisor graph, essential graph, reduced ring 13A15; 05C99; 05C25. Let \(A\) be a commutative ring with non-zero identity, and \(E(A)=\{p\in A | ann_A(pq)\leq_e A, ~\mbox { for some } ~q\in A^* \}\) . The extended essential graph, denoted by \(E_gG(A)\) is a graph with the vertex set \(E(A)^*=E(A)\setminus\{0\}\). Two distinct vertices \(r, s\in E(A)^*\) are adjacent if and only if \(ann_A(rs)\leq_e A\). In this paper, we prove that \(E_gG(A)\) is connected with \(diam(E_gG(A))\leq 3\) and if \(E_gG(A)\) has a cycle, then \(gr(E_gG(A))\leq 4\). Furthermore, we establish that if \(A\) is an Artinian commutative ring, then \(\omega (E_gG(A))=\chi (E_gG(A))=|N(A)^*|+ |Max(A)|\). Lugansk National Taras Shevchenko University 2024-04-21 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2120 10.12958/adm2120 Algebra and Discrete Mathematics; Vol 37, No 1 (2024) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2120/pdf Copyright (c) 2024 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2024-04-21T17:47:57Z |
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OJS |
| language |
English |
| topic |
zero-divisor graph essential graph reduced ring 13A15 05C99 05C25. |
| spellingShingle |
zero-divisor graph essential graph reduced ring 13A15 05C99 05C25. Ali, Asma Ahmad, Bakhtiyar An extension of the essential graph of a ring |
| topic_facet |
zero-divisor graph essential graph reduced ring 13A15 05C99 05C25. |
| format |
Article |
| author |
Ali, Asma Ahmad, Bakhtiyar |
| author_facet |
Ali, Asma Ahmad, Bakhtiyar |
| author_sort |
Ali, Asma |
| title |
An extension of the essential graph of a ring |
| title_short |
An extension of the essential graph of a ring |
| title_full |
An extension of the essential graph of a ring |
| title_fullStr |
An extension of the essential graph of a ring |
| title_full_unstemmed |
An extension of the essential graph of a ring |
| title_sort |
extension of the essential graph of a ring |
| description |
Let \(A\) be a commutative ring with non-zero identity, and \(E(A)=\{p\in A | ann_A(pq)\leq_e A, ~\mbox { for some } ~q\in A^* \}\) . The extended essential graph, denoted by \(E_gG(A)\) is a graph with the vertex set \(E(A)^*=E(A)\setminus\{0\}\). Two distinct vertices \(r, s\in E(A)^*\) are adjacent if and only if \(ann_A(rs)\leq_e A\). In this paper, we prove that \(E_gG(A)\) is connected with \(diam(E_gG(A))\leq 3\) and if \(E_gG(A)\) has a cycle, then \(gr(E_gG(A))\leq 4\). Furthermore, we establish that if \(A\) is an Artinian commutative ring, then \(\omega (E_gG(A))=\chi (E_gG(A))=|N(A)^*|+ |Max(A)|\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2024 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2120 |
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