Computing h^*-polynomials of edge polytopes using visibility-activities

Edge polytopes are lattice polytopes defined for directed graphs, and their geometry gives a lot of information on the graph structure.

These polytopes have nice unimodular triangulations that correspond to subsets of the spanning trees. We give a general method to compute the $h^*-$polynomial (a polynomial-valued refinement of the volume) based on such a triangulation, or even a dissection into interior disjoint simplices. Then, we give some concrete applications of our method that produce various graph-theoretic activity-formulas for the $h^*-$polynomial.

Joint work with Tamás Kálmán.