On the Ehrhart Theory of Generalized Symmetric Edge Polytopes

TL;DR

This work extends the Ehrhart theory of symmetric edge polytopes to generalized SEPs arising from regular matroids, employing combinatorial and Gröbner-basis techniques to transfer properties from ordinary SEPs and to reveal limitations via a counterexample to γ-nonnegativity. It introduces a near-γ-nonnegativity phenomenon by deleting two matroid elements, derives an explicit h*-polynomial formula for a key family Σ(Γ(n+1)), and provides a γ-vector expression showing γ-nonnegativity for that family. The results reinforce Ohsugi and Tsuchiya's conjecture in the ordinary SEP case while highlighting new complexity in the generalized setting. The paper also develops a framework connecting triangulations, toric ideals, and matroid operations to compute Ehrhart invariants in this context.

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, Tóthmérész and, independently, D'Alí, 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 $γ$-vectors. In this article, we use combinatorial and Gröbner basis techniques to extend additional known properties of SEPs to generalized SEPs. Along the way, we show that generalized SEPs are not necessarily $γ$-nonnegative by providing explicit examples. We prove that the polytopes we construct are ``nearly'' $γ$-nonnegative in the sense that, by deleting exactly two elements from the matroid, one obtains SEPs for graphs that are $γ$-nonnegative. This provides further evidence that Ohsugi and Tsuchiya's conjecture holds in the ordinary case.

Figures (5)

• Figure 1: The graphs $\Gamma(3)$, left, and $\Gamma(4)$, right.
• Figure 2: On the left, $K_{3,4}$ with a choice of $e$ and $e'$ labeled such that $\Gamma(4)$ is an underlying graph of $M^*(K_{3,4}) - \{e,e'\}$. On the right, a planar representation of $K_{3,4}/\{e,e'\}$ in solid black lines, with its dual, $\Gamma(4)$, overlaid in dashed gray lines.
• Figure 3: A spanning tree of $\Gamma(5)$, indicated by the solid edges. The edges $u_1v_1$ and $v_1u_2$ form a chordless pair. The edges $u_2v_2$ and $u_2u_3$ form a chorded pair of type $\alpha$, while the edges $u_4v_4$ and $u_4u_5$ form a chorded pair of type $\beta$. The edge $v_3u_4$ is an unpaired edge.
• Figure 4: Two spanning trees of $\Gamma(4)$ that are members of the triangulating tree set, as described in Proposition \ref{['prop: spanning tree characterization']}.
• Figure 5: Examples of modified spanning trees in $\Gamma(4)$. The original spanning trees and their corresponding modified trees are placed above and below each other.

Theorems & Definitions (41)

• Theorem 2.1: DJKK
• Theorem 2.2: KalmanTothmeresz
• Lemma 2.3
• Proposition 3.1: cf. DJKK
• Proposition 3.2: Braunfreesum
• Corollary 3.3
• Proposition 3.4: cf. OhsugiTsuchiya21
• proof
• Theorem 3.5: cf. DDM
• proof