Proof of a conjecture on graph polytope

Abstract

The graph polytopes arising from the vertex weighted graph, which was first introduced and studied by B\'ona, Ju, and Yoshida. A conjecture states that for a simple connected graph, the polynomial in the numerator of the Ehrhart series is palindromic. We confirm the conjecture. Furthermore, we introduce the hypergraph polytope. We prove that the simple connected unimodular hypergraph polytopes are integer polytopes. We also prove the polynomial in the numerator of the Ehrhart series of simple connected uniform hypergraph polytopes is palindromic.