On the Ehrhart Theory of Generalized Symmetric Edge Polytopes

TL;DR

This paper extends known properties of symmetric edge polytopes to their generalized versions for regular matroids, using combinatorial and algebraic techniques, and investigates their Ehrhart $ extit{h}^*$-polynomials and $ extit{ extgamma}$-nonnegativity.

Contribution

It generalizes properties of symmetric edge polytopes to regular matroids and explores $ extit{ extgamma}$-nonnegativity, providing explicit counterexamples and near-positivity results.

Findings

Generalized SEPs are not necessarily $ extgamma$-nonnegative.

By removing two elements from the matroid, the resulting SEP is $ extgamma$-nonnegative.

Supports the conjecture that ordinary SEPs have nonnegative $ extgamma$-vectors.

Abstract

The symmetric edge polytope (SEP) of a (finite, undirected) graph is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. SEPs have been studied extensively in the past twenty years. Recently, T\'othm\'er\'esz and, independently, D'Al\'i, Juhnke-Kubitzke, and Koch generalized the definition of an SEP to regular matroids, which are the matroids that can be represented by totally unimodular matrices. Generalized SEPs are known to have symmetric Ehrhart $h^*$-polynomials, and Ohsugi and Tsuchiya conjectured that (ordinary) SEPs have nonnegative $\gamma$-vectors. In this article, we use combinatorial and Gr\"obner basis techniques to extend additional known properties of SEPs to generalized SEPs. Along the way, we show that generalized SEPs are not necessarily $\gamma$-nonnegative by providing explicit examples. We prove that the polytopes we construct are ``nearly'' $\gamma$-nonnegative in the sense that, by deleting exactly two elements from the matroid, one obtains SEPs for graphs that are $\gamma$-nonnegative. This provides further evidence that Ohsugi and Tsuchiya's conjecture holds in the ordinary case.