a graph associated to especial essentiality of submodules

‎let $r$ be an associative ring with identity‎. ‎in this paper we‎‎associate to every $r$-module $m$ a simple graph $gamma_e(m)$‎, which we call it the essentiality graph of $m$. the vertices of $gamma_e(m)$ are nonzero submodules of $m$ and two distinct‎‎vertices $k$ and $l$ are considered to be adjacent if and only‎‎if $kcap l$ 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 of‎‎$gamma_e(m)$‎. ‎in general‎, ‎this graph is not connected‎. ‎we‎‎study some special cases in which $gamma_e(m)$ is‎‎complete or a union of complete connected components and give some examples illustrating each specific case‎.