Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proof of a conjecture on graph polytope

Feihu Liu

TL;DR

The paper proves that for every finite simple connected graph $G$, the numerator $H(x)$ of the Ehrhart series $\mathrm{Ehr}(P(G),x)$ is palindromic, using Stanley's reciprocity. It then extends the method to hypergraph polytopes, showing that unimodular simple connected hypergraph polytopes are integral and that, for simple connected uniform hypergraphs, the Ehrhart numerator is palindromic with degree linked to the uniformity. The results include explicit forms for the Ehrhart series and recover known cases such as circular graphs and hypercubes. By connecting Stanley reciprocity with reflexive-polytopes theory via Hibi's palindromic theorem, the work broadens the scope of symmetry phenomena in Ehrhart theory for graph- and hypergraph-derived polytopes.

Abstract

Graph polytopes arising from vertex-weighted graphs were first introduced by Bóna, 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.

Proof of a conjecture on graph polytope

TL;DR

The paper proves that for every finite simple connected graph , the numerator of the Ehrhart series is palindromic, using Stanley's reciprocity. It then extends the method to hypergraph polytopes, showing that unimodular simple connected hypergraph polytopes are integral and that, for simple connected uniform hypergraphs, the Ehrhart numerator is palindromic with degree linked to the uniformity. The results include explicit forms for the Ehrhart series and recover known cases such as circular graphs and hypercubes. By connecting Stanley reciprocity with reflexive-polytopes theory via Hibi's palindromic theorem, the work broadens the scope of symmetry phenomena in Ehrhart theory for graph- and hypergraph-derived polytopes.

Abstract

Graph polytopes arising from vertex-weighted graphs were first introduced by Bóna, 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.
Paper Structure (4 sections, 8 theorems, 39 equations, 4 figures)

This paper contains 4 sections, 8 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Graph Polytope
  3. Hypergraph Polytope
  4. Concluding Remark

Key Result

Lemma 1.1

Ehrhart62 Let $\mathcal{P}$ be a rational polytope of dimension $d$ in $\mathbb{R}^m$ with vertex set $\widehat{V}$. The Ehrhart series $\mathrm{Ehr}(\mathcal{P},x)$ is a rational function $\frac{R(x)}{T(x)}$ where $R(x)$ and $T(x)$ are polynomials satisfying:

Figures (4)

  • Figure 1: A unimodular hypergraph with $3$ hyperedges
  • Figure 2: A unimodular hypergraph with $5$ hyperedges
  • Figure 3: A $3$-uniform hypergraph with $3$ hyperedges
  • Figure 4: A $5$-uniform hypergraph with $4$ hyperedges

Theorems & Definitions (18)

  • Lemma 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: Stanley's reciprocity theorem, StanleyMagic73
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4: First proof in LeeJu15
  • ...and 8 more