  Let R be a commutative ring with unity, M be an unitary R-module and {\Gamma}
be a simple graph. This research article is an interplay of combinatorial and
algebraic properties of M . We show a combinatorial object completely
determines an algebraic object and characterise all finite abelian groups. We
discuss the correspondence between essential ideals of R, submodules of M and
vertices of graphs arising from M . We examine various types of equivalence
relations on objects of M. We study essential ideals corresponding to elements
of an object over hereditary and regular rings. Further, we study isomorphism
of annihilating graphs arising from M and tensor product.
