Perfectness of the essential graph for modules over commutative rings
Let $R$ be a commutative ring and $M$ be an $R$-module. The essential graph of $M$, denoted by $EG(M)$ is a simple graph with vertex set $Z(M) \setminus\operatorname{Ann}(M)$ and two distinct vertices $x,y \in Z(M) \setminus \operatorname{Ann}(M)$ are adjacent if and only if $\operatorname{Ann}_M(xy...
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Amirkabir University of Technology
2025-02-01
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| Series: | AUT Journal of Mathematics and Computing |
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| Online Access: | https://ajmc.aut.ac.ir/article_5327_0630d68c5c10927c46e609e527ddcb78.pdf |
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| author | Fatemeh Soheilnia Shiroyeh Payrovi Ali Behtoei |
| author_facet | Fatemeh Soheilnia Shiroyeh Payrovi Ali Behtoei |
| author_sort | Fatemeh Soheilnia |
| collection | DOAJ |
| description | Let $R$ be a commutative ring and $M$ be an $R$-module. The essential graph of $M$, denoted by $EG(M)$ is a simple graph with vertex set $Z(M) \setminus\operatorname{Ann}(M)$ and two distinct vertices $x,y \in Z(M) \setminus \operatorname{Ann}(M)$ are adjacent if and only if $\operatorname{Ann}_M(xy)$ is an essential submodule of $M$. In this paper, we investigate the dominating set, the clique and the chromatic number and the metric dimension of the essential graph for Noetherian modules. Let $M$ be a Noetherian $R$-module such that ${}|{} {\rm MinAss}_R(M){}|{}=n\geq 2$ and let $EG(M)$ be a connected graph. We prove that $EG(M)$ is a weakly prefect, that is, $\omega(EG(M))=\chi(EG(M))$. Furthermore, it is shown that $\dim (EG(M))= {}|{} Z(M){}|{}-({}|{}\operatorname{Ann}(M){}|{}+2^n)$, whenever $r(\operatorname{Ann}(M) )\not=\operatorname{Ann}(M)$ and $\dim (EG(M))= {}|{} Z(M){}|{}-({}|{}\operatorname{Ann}(M){}|{}+2^n-2)$, whenever $r(\operatorname{Ann}(M) )=\operatorname{Ann}(M)$. |
| format | Article |
| id | doaj-art-8dc9d97d177044319a131b12e15cbfab |
| institution | Kabale University |
| issn | 2783-2449 2783-2287 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | Amirkabir University of Technology |
| record_format | Article |
| series | AUT Journal of Mathematics and Computing |
| spelling | doaj-art-8dc9d97d177044319a131b12e15cbfab2025-02-11T12:37:04ZengAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24492783-22872025-02-016216317010.22060/ajmc.2023.22138.11365327Perfectness of the essential graph for modules over commutative ringsFatemeh Soheilnia0Shiroyeh Payrovi1Ali Behtoei2Department of Pure Mathematics, Faculty of Science, Imam Khomieini International University, Qazvin, IranDepartment of Pure Mathematics, Faculty of Science, Imam Khomieini International University, Qazvin, IranDepartment of Pure Mathematics, Faculty of Science, Imam Khomieini International University, Qazvin, IranLet $R$ be a commutative ring and $M$ be an $R$-module. The essential graph of $M$, denoted by $EG(M)$ is a simple graph with vertex set $Z(M) \setminus\operatorname{Ann}(M)$ and two distinct vertices $x,y \in Z(M) \setminus \operatorname{Ann}(M)$ are adjacent if and only if $\operatorname{Ann}_M(xy)$ is an essential submodule of $M$. In this paper, we investigate the dominating set, the clique and the chromatic number and the metric dimension of the essential graph for Noetherian modules. Let $M$ be a Noetherian $R$-module such that ${}|{} {\rm MinAss}_R(M){}|{}=n\geq 2$ and let $EG(M)$ be a connected graph. We prove that $EG(M)$ is a weakly prefect, that is, $\omega(EG(M))=\chi(EG(M))$. Furthermore, it is shown that $\dim (EG(M))= {}|{} Z(M){}|{}-({}|{}\operatorname{Ann}(M){}|{}+2^n)$, whenever $r(\operatorname{Ann}(M) )\not=\operatorname{Ann}(M)$ and $\dim (EG(M))= {}|{} Z(M){}|{}-({}|{}\operatorname{Ann}(M){}|{}+2^n-2)$, whenever $r(\operatorname{Ann}(M) )=\operatorname{Ann}(M)$.https://ajmc.aut.ac.ir/article_5327_0630d68c5c10927c46e609e527ddcb78.pdfessential graph dominating setclique number chromatic numbermetric dimension |
| spellingShingle | Fatemeh Soheilnia Shiroyeh Payrovi Ali Behtoei Perfectness of the essential graph for modules over commutative rings AUT Journal of Mathematics and Computing essential graph dominating set clique number chromatic number metric dimension |
| title | Perfectness of the essential graph for modules over commutative rings |
| title_full | Perfectness of the essential graph for modules over commutative rings |
| title_fullStr | Perfectness of the essential graph for modules over commutative rings |
| title_full_unstemmed | Perfectness of the essential graph for modules over commutative rings |
| title_short | Perfectness of the essential graph for modules over commutative rings |
| title_sort | perfectness of the essential graph for modules over commutative rings |
| topic | essential graph dominating set clique number chromatic number metric dimension |
| url | https://ajmc.aut.ac.ir/article_5327_0630d68c5c10927c46e609e527ddcb78.pdf |
| work_keys_str_mv | AT fatemehsoheilnia perfectnessoftheessentialgraphformodulesovercommutativerings AT shiroyehpayrovi perfectnessoftheessentialgraphformodulesovercommutativerings AT alibehtoei perfectnessoftheessentialgraphformodulesovercommutativerings |