en تخصصي Special پژوهشي Research paper Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete, tree or complete bipartite are studied and several characterizations are given. Associated Graph of module, Prime spectrum, Connected graph, Diameter. 45 58 http://ijmsi.ir/browse.php?a_code=A-10-438-1&slc_lang=en&sid=1 A. Abbasi aabbasi@guilan.ac.ir 10031947532846006252 10031947532846006252 Yes Uni. Guilan H. Roshan-Shekalgourabi hroshan@guilan.ac.ir 10031947532846006253 10031947532846006253 No Uni. Guilan D. Hassanzadeh-Lelekaami dhmath@guilan.ac.ir 10031947532846006254 10031947532846006254 No Uni. Guilan