Inspired by asymptotically flat manifolds, we introduce the concept of asymptotically flat graphs and define the discrete ADM mass on them. We formulate the discrete positive mass conjecture based on the scalar curvature in the sense of Ollivier curvature, and prove the positive mass theorem for asymptotically flat graphs that are combinatorially isomorphic to grid graphs. As a corollary, the discrete torus does not admit positive scalar curvature. We prove a weaker version of the positive mass conjecture: an asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph. Hence the combinatorial structure of an asymptotically flat graph is determined by the curvature condition, which is a discrete analog of the rigidity part for the positive mass theorem. The key tool for the proof is the discrete harmonic function of linear growth associated with the salami structure.