  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
$\mathbb{Z}$-graded 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.
