We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm to efficiently compute reduced divisors. Applications include an "efficient bijective" proof of Kirchhoff's matrix-tree theorem and a new algorithm for finding random spanning trees. The running times of our algorithms are analyzed using potential theory, and we show that the bounds thus obtained generalize and improve upon several previous results in the literature. We also extend some of these considerations to metric graphs.Comment: To appear in Journal of Combinatorial Theory, Series A -- Revised and updated. The discussion on metric graphs (now in Appendix A) will not appear in the journal version. Proofs of the Dhar theorem and the Cori-Le Borgne theorem are in v1 but not in v

Let Γ be a compact tropical curve (or metric graph) of genus g. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree g on Γ . We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an 'integral' version of this result which is of independent interest. As an application, we provide a 'geometric proof' of (a dual version of) Kirchhoff's celebrated matrixtree theorem. Indeed, we show that each weighted graph model G for Γ gives rise to a canonical polyhedral decomposition of the g-dimensional real torus Pic g (Γ ) into parallelotopes C T , one for each spanning tree T of G, and the dual Kirchhoff theorem becomes the statement that the volume of Pic g (Γ ) is the sum of the volumes of the cells in the decomposition.

We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud--Bosch--L\"utkebohmert theory of non-Archimedean theta functions for abelian varieties with semi-abelian reduction.Comment: Final version to appear in Mathematische Annalen. Full-text view-only journal version is available at http://rdcu.be/E5tM . Note the journal version has a slightly different numbering syste

Abstract. We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their Zgraded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.

Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves.