Roots of Ehrhart polynomials arising from graphs

TL;DR

This paper investigates the roots of Ehrhart polynomials from graph-derived polytopes, supporting existing conjectures and proposing new ones about their distribution, with computational and theoretical insights into their locations.

Contribution

It introduces two new conjectures on the roots of Ehrhart polynomials for specific graph polytopes and provides supporting computational and theoretical evidence.

Findings

Roots of Ehrhart polynomials of complete multipartite graph edge polytopes lie in a specific circle or are negative integers.

Ehrhart polynomial roots of Gorenstein Fano polytopes are confined within a narrower vertical strip.

Computational plots illustrate the root distributions for various polytopes.

Abstract

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart polynomials of polytopes of dimension $D$ satisfy $-D \le \Re(\alpha) \le D - 1$, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order $d$ lie in the circle $|z+\tfrac{d}{4}| \le \tfrac{d}{4}$ or are negative integers, and (2) a Gorenstein Fano polytope of dimension $D$ has the roots of its Ehrhart polynomial in the narrower strip $-\tfrac{D}{2} \leq \Re(\alpha) \leq \tfrac{D}{2}-1$. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.