Ehrhart theory, Modular flow reciprocity, and the Tutte polynomial

TL;DR

This paper links the modular flow polynomial of an oriented graph to Ehrhart polynomials of lattice polytopes, providing new combinatorial interpretations and extending reciprocity theorems to Tutte polynomial evaluations.

Contribution

It introduces a novel geometric perspective on modular flow polynomials using Ehrhart theory and extends reciprocity results to Tutte polynomial evaluations.

Findings

Provides a combinatorial interpretation for modular flow polynomial at negative arguments.

Extends Ehrhart-Macdonald reciprocity to graph flows and tensions.

Offers an enumerative interpretation for positive Tutte polynomial evaluations.

Abstract

Given an oriented graph G, the modular flow polynomial 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 the modular flow polynomial at negative arguments which answers a question of Beck and Zaslavsky (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 of G.