Directed graphs and lattice polytopes

I will survey recent results, all joint with Lilla Tóthmérész, on lattice polytopes associated to graphs and their h*-polynomials. More precisely, any directed graph has a natural root (a.k.a. edge) polytope, which we extend to always include the origin; when the graph is bidirected, this is known as a symmetric edge polytope. I will present formulas for the degree of the corresponding h*-polynomial. Moreover, it turns out that if any edge of the digraph is either deleted or contracted to a point, then no coefficient in the polynomial will increase. Curiously, many cases can be found in which the contraction of an edge does not change the polynomial, even as the dimension of the polytope reduces. In the case of digraphs with a Gorenstein polytope (i.e., a palindromic h*-polynomial), this provides examples of the phenomenon, due to Batyrev and Nill, that such polytopes can be projected along a special simplex onto a reflexive polytope with the same h*-polynomial. If time permits then I will also discuss comparisons, mostly conjectural, between the h*-polynomials derived from various orientations of the same graph.