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 $Z(M) \setminus {\rm{Ann}}_R(M)$ and two distinct vertices $x,y \in Z(M) \setminus {\rm{Ann}}_R(M)$ are adjacent if and only if ${\rm{Ann}}_M(xy)$ is an essential submodule of $M$. In this paper, we investigate the dominating set, the clique and the chromatic numbers and the metric dimension of the essential graph for Noetherian modules. Let $M$ be a Noetherian $R$-module such that $\mid {\rm MinAss}_R(M)\mid=n\geq 2$ and let $EG(M)$ be a connected graph. We prove that $EG(M)$ is weakly prefect, that is, $\omega(EG(M))=\chi(EG(M))$. Furthermore, it is shown that ${\rm dim}(EG(M))=\mid Z(M)\mid-(\mid {\rm{Ann}}(M)\mid+2^n)$, whenever $r({\rm{Ann}}(M) )\neq{\rm{Ann}}(M)$ and ${\rm dim}(EG(M))=\mid Z(M)\mid-(\mid {\rm{Ann}}(M)\mid+2^n-2)$, whenever $r({\rm{Ann}}(M) )= {\rm{Ann}}(M)$.
Payrovi, S., Soheilnia, F., & Behtoei, A. (2023). Perfectness of the essential graph for modules over commutative rings. AUT Journal of Mathematics and Computing, (), -. doi: 10.22060/ajmc.2023.22138.1136
MLA
Shiroyeh Payrovi; Fatemeh Soheilnia; Ali Behtoei. "Perfectness of the essential graph for modules over commutative rings". AUT Journal of Mathematics and Computing, , , 2023, -. doi: 10.22060/ajmc.2023.22138.1136
HARVARD
Payrovi, S., Soheilnia, F., Behtoei, A. (2023). 'Perfectness of the essential graph for modules over commutative rings', AUT Journal of Mathematics and Computing, (), pp. -. doi: 10.22060/ajmc.2023.22138.1136
VANCOUVER
Payrovi, S., Soheilnia, F., Behtoei, A. Perfectness of the essential graph for modules over commutative rings. AUT Journal of Mathematics and Computing, 2023; (): -. doi: 10.22060/ajmc.2023.22138.1136
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