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The symmetric edge polytope (SEP) of a (finite, undirected) graph is acentrally symmetric lattice polytope whose vertices are defined by the edges ofthe graph. SEPs have been studied extensively in the past twenty years.Recently, D'Ali, Juhnke-Kubitzke, and Koch generalized the definition of an SEPto regular matroids, as these are the matroids that can be characterized bytotally unimodular matrices. Generalized SEPs are known to have symmetricEhrhart $h^*$-polynomials, and Ohsugi and Tsuchiya conjectured that (ordinary)SEPs have nonnegative $\gamma$-vectors.  In this article, we use combinatorial and Gr\"obner basis techniques toextend additional known properties of SEPs to generalized SEPs. Along the way,we show that generalized SEPs are not necessarily $\gamma$-nonnegative byproviding explicit examples. We conjecture these polytopes to be ``nearly''$\gamma$-nonnegative in the sense that, by deleting exactly two elements fromthe matroid, one obtains SEPs for graphs that are $\gamma$-nonnegative. Thisprovides further evidence that Ohsugi and Tsuchiya's conjecture holds in theordinary case.

On the Ehrhart Theory of Generalized Symmetric Edge Polytopes