Bounded powers of edge ideals: Nearly Gorenstein polytopes

A lattice polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called nearly Gorenstein if $\mathcal{P}$ is normal and if its Ehrhart ring is a nearly Gorenstein ring. We study nearly Gorenstein polytopes arising from discrete polymatroids of bounded powers of edge ideals. Especially, it is shown that every nearly Gorenstein polytope coming from either a complete graph or a complete bipartite graph is level.