Minimal submodules graph of modules over commutative rings

Let R be a commutative ring and M be an R-module. In this paper, we define minimal submodules graph of M, denoted by Γₘᵢₙ(M), in which the vertex set is the set of nonzero proper submodules of M. Two distinct vertices A and B are adjacent provided that A∩ B is a minimal submodule of M. In this study, we associate some properties of the graph from the properties of module and vice versa. Moreover, if we have an R-module homomorphism from M to $M'$, we compare some invariant numbers and properties of Γₘᵢₙ(M) and Γₘᵢₙ(M').