Parole chiave

**Abstract** In this talk I will focus on two integral convex polytopes one can construct from a graph: symmetric edge polytopes and cosmological polytopes. Both constructions have been introduced and studied in the context of physics, and the former plays a key role also in the study of finite metric spaces and optimal transport. The general goal is to understand how interesting geometric invariants of these polytopes (their Ehrhart theory, volume, triangulations) are related to the combinatorics of the underlying graph. In the first part I will present a conjecture due to Ohsugi and Tsuchiya on a numerical invariant called the $h^\ast$-vector of symmetric edge polytopes, and present partial results obtained in a joint work with Alessio D’Alì, Martina Juhnke-Kubitzke and Daniel Köhne. In the second part I will focus on cosmological polytopes. In a joint work (in progress) with Martina Juhnke-Kubitzke and Liam Solus we show that all these polytopes have a regular unimodular triangulation. As a result we compute the volume of the cosmological polytope of the cycle graph.