Two-Variable Zeta-Functions on Graphs and Riemann–Roch Theorems

We investigate, in this article, a generalization of the Riemann–Roch theorem for graphs of Baker and Norine, with a view toward identifying new objects for which a two-variable zeta-function can be defined. To a lattice Λ of rank n−1 in $\mathbb Z^n$ and perpendicular to a positive integer vector R, we define the notions of g-number and of canonical vector, in analogy with the notions of genus and canonical class in the theory of algebraic curves. When Λ is the full sublattice of $\mathbb Z^n$ perpendicular to R, its g-number turns out to be the classical Frobenius number of the coefficients of R. We investigate the existence of canonical vectors—in particular, in the context of arithmetical graphs—where we obtain an existence theorem using methods from arithmetic geometry. We show that a two-variable zeta-function can be defined when a canonical vector exists.