Linear systems and Riemann-Roch theory on graphs

Graphs can be viewed as discrete counterparts to algebraic curves, as exemplified by the recent Riemann-Roch formula for integral divisors on multigraphs. We show that for any subring R of the reals, the Riemann-Roch formula can be generalized to R-valued divisors on edge-weighted graphs over R. We also show that a related abelian sandpile model extended to R on edge-weighted graphs leads to a group, which has many interesting properties. The sandpile results are used to prove various properties of linear systems of divisors on graphs, including that the set of divisors with empty linear systems is completely determined by a lattice of nonspecial divisors. We use these properties of linear systems on graphs to study line bundles on binary and ternary algebraic curves that match the dimension of their graph counterparts.