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Number of facets of symmetric edge polytopes arising from join graphs

Aki Mori, Kenta Mori, Hidefumi Ohsugi

TL;DR

This work analyzes the facet structure of symmetric edge polytopes ${\mathcal{P}}_G$ and establishes sharp bounds on their number of facets for specific graph constructions. By leveraging facet-subgraph decompositions, dominated-set characterizations for suspension graphs, and a detailed partitioning argument for join graphs, the authors prove Braun–Bruegge's conjecture for join graphs and connect these results to Nill's reflexive-polytopes bound $N\le 6^{d/2}$. The main contributions include a complete bound for suspension graphs and a rigorous extension to join graphs, with precise equality cases that identify the extremal graph configurations. This advances understanding of how graph operations influence facet counts in symmetric edge polytopes and informs related questions in Ehrhart theory and toric geometry.

Abstract

Symmetric edge polytopes of graphs are important object in Ehrhart theory,and have an application to Kuramoto models. In the present paper, we study the upper and lower bounds for the number of facets of symmetric edge polytopes of connected graphs conjectured by Braun and Bruegge. In particular, we show that their conjecture is true for any graph that is the join of two graphs (equivalently, for any connected graph whose complement graph is not connected). It is known that any symmetric edge polytope is a centrally symmetric reflexive polytope. Hence our results give a partial answer to Nill's conjecture: the number of facets of a $d$-dimensional reflexive polytope is at most $6^{d/2}$.

Number of facets of symmetric edge polytopes arising from join graphs

TL;DR

This work analyzes the facet structure of symmetric edge polytopes and establishes sharp bounds on their number of facets for specific graph constructions. By leveraging facet-subgraph decompositions, dominated-set characterizations for suspension graphs, and a detailed partitioning argument for join graphs, the authors prove Braun–Bruegge's conjecture for join graphs and connect these results to Nill's reflexive-polytopes bound . The main contributions include a complete bound for suspension graphs and a rigorous extension to join graphs, with precise equality cases that identify the extremal graph configurations. This advances understanding of how graph operations influence facet counts in symmetric edge polytopes and informs related questions in Ehrhart theory and toric geometry.

Abstract

Symmetric edge polytopes of graphs are important object in Ehrhart theory,and have an application to Kuramoto models. In the present paper, we study the upper and lower bounds for the number of facets of symmetric edge polytopes of connected graphs conjectured by Braun and Bruegge. In particular, we show that their conjecture is true for any graph that is the join of two graphs (equivalently, for any connected graph whose complement graph is not connected). It is known that any symmetric edge polytope is a centrally symmetric reflexive polytope. Hence our results give a partial answer to Nill's conjecture: the number of facets of a -dimensional reflexive polytope is at most .
Paper Structure (4 sections, 16 theorems, 55 equations, 3 figures)

This paper contains 4 sections, 16 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basics on the facets of symmetric edge polytopes
  3. Facets of symmetric edge polytopes of suspension graphs
  4. Join graphs

Key Result

Theorem 1.3

Let $G$ be a graph on the vertex set $[n-1]$ with $n \ge 2$. Then and equality holds if and only if $G$ is an empty graph (i.e., a graph having no edges), and hence ${\widehat{G}}$ is a star graph $K_{1,n-1}$. Moreover, and equality holds if and only if one of the following holds :

Figures (3)

  • Figure 1: Example of $\mu(H)$ from Lemma \ref{['countlemma']}.
  • Figure 2: Three graphs obtained from $G$ and $v$.
  • Figure :

Theorems & Definitions (35)

  • Conjecture 1.1: NillConj
  • Conjecture 1.2: BrBr
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1: HJM
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4: CDK
  • Example 2.5
  • Proposition 2.6: DDM
  • ...and 25 more