Number of facets of symmetric edge polytopes arising from join graphs

TL;DR

This paper investigates the bounds on the number of facets of symmetric edge polytopes derived from join graphs, confirming a conjecture for a broad class of graphs and contributing to understanding reflexive polytopes.

Contribution

It proves the Braun and Bruegge conjecture for join graphs, linking graph structure to polytope facets and partially addressing Nill's conjecture on reflexive polytopes.

Findings

Conjecture holds for join graphs and graphs with disconnected complements.

Number of facets bounded by 6^{d/2} for d-dimensional reflexive polytopes.

Provides new bounds and structural insights into symmetric edge polytopes.

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 $6^{d/2}$.