On the gamma-vector of symmetric edge polytopes

TL;DR

This paper investigates the properties of gamma-vectors of symmetric edge polytopes, proving nonnegativity of certain entries, characterizing cases of equality, and showing asymptotic nonnegativity for random graphs, thus supporting Gal's conjecture.

Contribution

It proves nonnegativity of gamma_2 for all graphs, characterizes when gamma_2 is zero, and demonstrates asymptotic nonnegativity for gamma-vectors of random graphs, confirming Gal's conjecture in this context.

Findings

Gamma_2 is nonnegative for any graph.

Complete characterization of when gamma_2 equals zero.

Gamma-vectors of most Erdős-Rényi random graphs are asymptotically nonnegative.

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.