Viewing counting polynomials as Hilbert functions via Ehrhart theory

TL;DR

This paper links counting polynomials in graph theory to Hilbert functions using Ehrhart theory, providing criteria for when these polynomials are Hilbert functions and applying this to flow and tension polynomials.

Contribution

It introduces a criterion connecting Ehrhart polynomials of polytopal complexes to Hilbert functions and applies it to various graph polynomials.

Findings

Flow and tension polynomials are Hilbert functions under certain conditions

Ehrhart theory provides a new perspective on counting polynomials

A sufficient criterion for Ehrhart polynomials to be Hilbert functions

Abstract

Steingrimsson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative Stanley-Reisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the Ehrhart polynomial of a given relative polytopal complex is a Hilbert function in Steingrimsson's sense. We use this result to establish that the modular and integral flow and tension polynomials of a graph are Hilbert functions.