Line graph of extensions of the zero-divisor graph in commutative rings
UDC 512.6 We consider a finite commutative ring with unity denoted by $\mathscr{P}.$ Within this framework, the essential graph of $\mathscr{P}$ is represented as $E{G}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set, and two distinct vertices $x$ and $y$ are a...
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| Date: | 2025 |
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| Main Authors: | , , |
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| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2025
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7817 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512764009644032 |
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| author | Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd |
| author_facet | Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd |
| author_sort | Rehman, Nadeem ur |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-08-09T14:51:13Z |
| description | UDC 512.6
We consider a finite commutative ring with unity denoted by $\mathscr{P}.$ Within this framework, the essential graph of $\mathscr{P}$ is represented as $E{G}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann(xy)$ is an essential ideal of $\mathscr{P}.$ At the same time, the weakly zero-divisor graph of $\mathscr{P}$ is denoted by $\text{W}{\Gamma}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set and an edge is defined between two distinct vertices $u$ and $v$ if and only if there exist $r \in ann(u)^*$ and $s\in ann(v)^*$ such that $rs=0$, where $ann(u) = \{v \in \mathscr{P}\colon uv = 0\}$ for $u \in \mathscr{P}.$ In our research, we deal with the conditions under which both $E{G}(\mathscr{P})$ and $\text{W}{\Gamma}(\mathscr{P})$ can be classified as line graphs. Furthermore, we explore the scenarios in which these graphs are the complements of line graphs. |
| doi_str_mv | 10.3842/umzh.v76i11.7817 |
| first_indexed | 2026-03-24T03:33:58Z |
| format | Article |
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| id | umjimathkievua-article-7817 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:33:58Z |
| publishDate | 2025 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-78172025-08-09T14:51:13Z Line graph of extensions of the zero-divisor graph in commutative rings Line graph of extensions of the zero-divisor graph in commutative rings Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd Weakly zero-divisor graph essential graph line graph complement of a line graph commutative rings UDC 512.6 We consider a finite commutative ring with unity denoted by $\mathscr{P}.$ Within this framework, the essential graph of $\mathscr{P}$ is represented as $E{G}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann(xy)$ is an essential ideal of $\mathscr{P}.$ At the same time, the weakly zero-divisor graph of $\mathscr{P}$ is denoted by $\text{W}{\Gamma}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set and an edge is defined between two distinct vertices $u$ and $v$ if and only if there exist $r \in ann(u)^*$ and $s\in ann(v)^*$ such that $rs=0$, where $ann(u) = \{v \in \mathscr{P}\colon uv = 0\}$ for $u \in \mathscr{P}.$ In our research, we deal with the conditions under which both $E{G}(\mathscr{P})$ and $\text{W}{\Gamma}(\mathscr{P})$ can be classified as line graphs. Furthermore, we explore the scenarios in which these graphs are the complements of line graphs. УДК 512.6 Лінійний граф iз розширень графа з нульовим дільником у комутативних кільцях Розглянуто скінченне комутативне кільце з одиницею, позначене як $\mathscr{P}.$ У цьому випадку для суттєвого графа $\mathscr{P}$ справедливим є зображення $E{G}(\mathscr{P})$ з множиною вершин $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\},$ у якій дві різні вершини $x$ та $y$ суміжні тоді й лише тоді, коли $ann(xy)$ є суттєвим ідеалом $\mathscr{P}.$ Водночас слабкий  граф із нульовим дільником $\mathscr{P}$  позначено як $\text{W}{\Gamma}(\mathscr{P})$ з множиною вершин $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\},$ де ребро визначено між двома різними вершинами $u$ та $v$ тоді й лише тоді, коли існують $r \in {\rm ann}(u)^*$ і $s\in ann(v)^*$ такі, що $rs=0$, де $ann(u) = \{v \in \mathscr{P}\colon uv = 0\}$ для $u \in \mathscr{P}.$ У цьому дослідженні розглянуто умови, за яких $E{G}(\mathscr{P})$ і $\text{W}{\Gamma}(\mathscr{P})$ можна класифікувати як лінійні графи. Крім того, досліджено сценарії, у яких ці графи є доповненнями лінійних графів.  Institute of Mathematics, NAS of Ukraine 2025-08-06 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7817 10.3842/umzh.v76i11.7817 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 11 (2024); 1645 - 1652 Український математичний журнал; Том 76 № 11 (2024); 1645 - 1652 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7817/10282 Copyright (c) 2024 Nadeem ur Rehman |
| spellingShingle | Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd Rehman, Nadeem ur Mir, Shabir Ahmad Nazim, Mohd Line graph of extensions of the zero-divisor graph in commutative rings |
| title | Line graph of extensions of the zero-divisor graph in commutative rings |
| title_alt | Line graph of extensions of the zero-divisor graph in commutative rings |
| title_full | Line graph of extensions of the zero-divisor graph in commutative rings |
| title_fullStr | Line graph of extensions of the zero-divisor graph in commutative rings |
| title_full_unstemmed | Line graph of extensions of the zero-divisor graph in commutative rings |
| title_short | Line graph of extensions of the zero-divisor graph in commutative rings |
| title_sort | line graph of extensions of the zero-divisor graph in commutative rings |
| topic_facet | Weakly zero-divisor graph essential graph line graph complement of a line graph commutative rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7817 |
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