We discuss natural binomial and monomial ideals related to the theory of divisors on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe minimal polyhedral cellular free resolutions for these ideals. 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.

This talk is based on joint works with F. Mohammadi. No advanced knowledge in commutative algebra will be assumed.