A proposal for research into the Jacobians of graphs

Abstract

Our potential theory methods allow us to prove some new results about chip-firing games and to give new proofs and/or generalizations of some known results in the subject. We also show that certain ``ad-hoc'' techniques in the literature are naturally explained or unified by our approach. In particular, we characterize reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and we provide an algorithm to efficiently compute reduced divisors.