Reciprocity and Polynomial Properties for Even Flows and Potentials on Directed Graphs

Abstract

We consider special types of mod-λ flows, called odd and even mod-λ flows, for directed graphs, and prove that the numbers of such flows can be interpolated by polynomials in λ with the degree given by the cycle rank of the graph. The proofs involve computation of the number of integer solutions in a polyhedral region of Euclidean space using theorems due to Ehrhart. The resulting reciprocity properties of the interpolating polynomials for even flows are considered. The analogous properties of odd and even mod-λ potential differences and their associated potentials are also developed.