ON THE GAMMA-VECTOR OF SYMMETRIC EDGE POLYTOPES

We study γ-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 γ2 for any graph and completely characterize the case when γ2 = 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 γ-vectors of symmetric edge polytopes of most Erdos-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.