The number of edges of a symmetric edge polytope

TL;DR

This paper establishes a sharp lower bound on the number of edges of symmetric edge polytopes derived from simple graphs, characterizes graphs that attain this bound, and explores related polynomial properties and edge-deletion effects.

Contribution

It provides the first sharp lower bound for edges of symmetric edge polytopes based on graph invariants and characterizes extremal graphs, connecting polytope geometry with graph theory.

Findings

Sharp lower bound for edges of symmetric edge polytopes

Characterization of graphs attaining the bound

Analysis of h*-polynomial behavior under edge deletion

Abstract

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.