Perfectness of the essential graph for modules over commutative rings - Fatemeh Soheilnia; Shiroyeh Payrovi; Ali Behtoei (AUT Journal of Mathematics and Computing)
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 ${}|{} {\rm MinAss}_R(M){}|{}=n≥ 2$ and let $EG(M)$ be a connected graph. We prove that $EG(M)$ is a weakly prefect, that is, $ω(EG(M))=χ(EG(M))$. 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 $r(\operatorname{Ann}(M) )=\operatorname{Ann}(M)$.