On the gamma-vector of symmetric edge polytopes

Abstract

We study $\gamma$-vectors associated with $h^{\ast }$-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, we prove nonnegativity of ${\gamma }_2$ for any graph and completely characterize the case when ${\gamma }_2=0$. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, we show that the $\gamma$-vectors of symmetric edge polytopes of most Erdős-Rényi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal's conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting.

arXiv commentv2: 30 pages, 4 figures. To appear on SIAM Journal on Discrete Mathematics

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Cited by in corpus (2)

• Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering
• Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs