Proof of a conjecture on graph polytope

TL;DR

This paper proves a conjecture that the Ehrhart series numerator polynomial of graph polytopes is palindromic for connected graphs, and extends results to hypergraph polytopes.

Contribution

It establishes the palindromic nature of Ehrhart series numerator polynomials for graph and hypergraph polytopes, and introduces hypergraph polytopes with new properties.

Findings

Numerator polynomial of Ehrhart series is palindromic for connected graphs.

Hypergraph polytopes are shown to be integer polytopes.

Palindromic Ehrhart series numerator extends to uniform hypergraph polytopes.

Abstract

Graph polytopes arising from vertex-weighted graphs were first introduced by B\'ona, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is palindromic, using Stanley's reciprocity theorem. Furthermore, we introduce hypergraph polytopes and establish that every simple, connected, unimodular hypergraph polytope is an integer polytope. Additionally, for simple connected uniform hypergraph polytopes, we demonstrate that the numerator polynomial of their Ehrhart series is palindromic.