Edges of symmetric edge polytopes and their Ehrhart theory

The symmetric edge polytope of a simple graph is a lattice polytope defined in terms of the edges of the graph. These polytopes appear in different contexts and recently have attracted attention as a family with interesting Ehrhart theoretic properties. We will discuss a sharp lower bound for the number of edges of symmetric edge polytopes, expressed only in terms of invariants of the graph. We then highlight a connection with the $h^∗-$polynomial of such polytopes, and in particular, discuss implications for their conjectured Γ-positivity.