Ehrhart theory, modular flow reciprocity, and the Tutte polynomial

Abstract

Given an oriented graph G, the modular flow polynomial φ G(k) counts the number of nowhere-zero ℤ k-flows of G. We give a description of the modular flow polynomial in terms of (open) Ehrhart polynomials of lattice polytopes. Using Ehrhart-Macdonald reciprocity we give a combinatorial interpretation for the values of φ G at negative arguments which answers a question of Beck and Zaslavsky (Adv Math 205:134-162, 2006). Our construction extends to ℤ l-tensions and we recover Stanley's reciprocity theorem for the chromatic polynomial. Combining the combinatorial reciprocity statements for flows and tensions, we give an enumerative interpretation for positive evaluations of the Tutte polynomial t G(x, y) of G. © 2010 The Author(s).