Orientations, lattice polytopes, and group arrangements II: Modular and integral flow polynomials of graphs

TL;DR

This paper explores the geometric and algebraic properties of modular and integral flow polynomials of graphs using subgroup arrangements and Ehrhart polynomials, revealing new reciprocity laws and interpretations.

Contribution

It introduces an Eulerian equivalence relation and applies Ehrhart polynomial theory to derive properties and reciprocity laws for flow polynomials in a geometric framework.

Findings

Established a reciprocity law for modular flow polynomials

Provided an interpretation of polynomial values at negative integers

Connected flow polynomials with Ehrhart theory and subgroup arrangements

Abstract

We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law for the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers, and answers a question by Beck and Zaslavsky.