Summary: In this paper, a new kind of graph on a unitary module \(A\) over a commutative ring \(R\) with identity, namely the co-maximal submodule graph is defined and studied as a natural generalization of the comaximal ideal graph of a commutative ring \(R\), denoted by \(\mathbb{C}(R)\). We use \(\mathbb{C}(A)\) to denote this graph, with its vertices the proper submodules of \(A\) which are not contained in the Jacobson radical of \(A\), and two vertices \(B_1\) and \(B_2\) are adjacent if and only if \(B_1+B_2 = A\). We show some properties of this graph and compare some of the results of \(\mathbb{C}(A)\) and \(\mathbb{C}(R)\). For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal submodules of the module \(A\). It is shown that \(\mathbb{C}(R)\) is isomorphic to a subgraph of \(\mathbb{C}(A)\) when \(A\) is a finitely generated cancellation (in particular, a finitely generated free) R-module. We also discuss the conditions under which \(A\) is a finite direct sum of simple modules, \(\mathbb{C}(A)\) is isomorphic to a finite Boolean graph, and \(\mathbb{C}(A)\) and \(\mathbb{C}(R)\) are isomorphic graphs.