We study square-free S-modules with support on a simplicial graph, and
investigate an analogy between Cohen--Macaulay modules, locally of rank 1,
supported on a connected graph and line bundles on a curve. We use the
combinatorial structure of the graph to prove a corresponding Riemann-Roch
theorem, we study the Jacobian for a connected graph, and we study
Brill-Noether theory for 2-connected graphs.