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) ann(m) and two distinct vertices x, y ∈ z(m) ann(m) are adjacent if and only if annm(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 |minassr(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)| − (| ann(m)| + 2n), whenever r(ann(m)) ̸= ann(m) and dim(eg(m)) = |z(m)| − (| ann(m)| + 2n − 2), whenever r(ann(m)) = ann(m)