  In this thesis we study toric rank functions for chip firing games and prove
special cases of a conjectural Riemann-Roch. The original motivation for an
investigation into this area of study came for the adaptation (due to Matt
Baker) of Riemann-Roch into a graph theoretic analogue through the use of
chip-firing games. Here, we collect known results and present new observations
that indicate Riemann--Roch holds for trees and polygons. We also prove an
asymptotic case of Riemann--Roch (i.e.~Riemann--Roch for divisors of large
degree). Finally, we also provide magma code and computational evidence that
Riemann--Roch holds for the toric rank function.
