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arXiv · 2201.09835

On the gamma-vector of symmetric edge polytopes

Alessio D'Alì·Martina Juhnke-Kubitzke·Daniel Köhne·Lorenzo Venturello
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Abstract

We study $\gamma$-vectors associated with $h^*$-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 Erd\H{o}s-R\'enyi 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.

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Alessio D'Alì, Martina Juhnke-Kubitzke, Daniel Köhne, Lorenzo Venturello. 2022-01-24. On the gamma-vector of symmetric edge polytopes. https://doi.org/10.1137/22m1492799

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