The number of edges of a symmetric edge polytope

The symmetric edge polytope of a simple graph is a lattice polytope defined as the convex hull of a subset of the type A roots corresponding to the edges of the graph.

In this article we prove a sharp lower bound for the number of edges of the symmetric edge polytope of a graph as a function of elementary graph invariants.

Moreover, we characterize graphs attaining this bound.

We highlight a connection with the h*-polynomial of such polytopes and, motivated by a conjecture of Ohsugi and Tsuchiya, we investigate the behaviour of such polynomial under edge-deletion in the graph.