On the Ehrhart Theory of Generalized Symmetric Edge Polytopes
arxiv(2024)
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, as these are the
matroids that can be characterized 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 these polytopes to be "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.
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