Frobenius Number, Riemann-roch Structure, and Zeta Functions of Graphs

Let R ∈ Z be a vector with strictly positive integers entries. We denote its transpose by R = (r1, . . . , rn). In this article, unless specified otherwise, any integer vector denoted R will be assumed to have gcd(r1, . . . , rn) = 1. Let D ∈ Z. We define the degree of D as degR(D) := D ·R. When the context makes the reference to R unnecessary, we will denote degR simply by deg. The kernel of the degree homomorphism Z → Z is the lattice in Z perpendicular to R: ΛR := {D ∈ Z, D ·R = 0}. For any sublattice Λ ⊆ ΛR of rank n − 1, we define Pic(Λ) := Z/Λ. If D ∈ Z, we denote by [D] the class of D in Pic(Λ). By construction, deg(Λ) = {0}, so that we have a group homomorphism