Riemann–Roch and Abel–Jacobi theory on a finite graph

It is well known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann–Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel–Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.