On the Gamma-Vector of Symmetric Edge Polytopes

We study-y-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-y2 for any graph and completely characterize the case when-y2 = 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-y-vectors of symmetric edge polytopes of most ErdoH \s--Re'\nyi 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.