Mehdi Ebrahimi Dorcheh - Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, P.O. Box ۶۵۷۱۹-۹۵۸۶۳, Malayer, Iran.
Saeid Bagheri - Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, P.O. Box ۶۵۷۱۹-۹۵۸۶۳, Malayer, Iran.

‎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 K\cap 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.

essential submodules, closed submodules, UC-module, Clique number, girth