A generalization of total graphs of modules

Let R be a commutative ring, and let M 6= 0 be an R-module with a non-zero proper submodule N, where N? = N − {0}. Let ΓN? (M) denote the (undirected) simple graph with vertices {x ∈ M − N | x + x 0 ∈ N? for some x 6= x 0 ∈ M − N}, where distinct vertices x and y are adjacent if and only if x + y ∈ N?. We determine some graph theoretic properties of ΓN? (M) and investigate the independence number and chromatic number.