Let R be an associative ring with identity. In this paper weassociate to every R-module M a simple graph ₑ (M) , which we call it the essentiality graph of M. The vertices of ₑ (M) are nonzero submodules of M and two distinctvertices K and L are considered to be adjacent if and onlyif K L is an essential submodule of K+L. We investigate the relationship between some module theoreticproperties of M such as minimality and closedness ofsubmodules with some graph theoretic properties ofₑ (M) . In general, this graph is not connected. Westudy some special cases in which ₑ (M) iscomplete or a union of complete connected components and give some examples illustrating each specific case.
Dorcheh et al. (Sun,) studied this question.