Facets of Symmetric Edge Polytopes for Graphs with Few Edges

TL;DR

This paper investigates the maximum and minimum number of facets of symmetric edge polytopes associated with graphs, providing formulas for sparse graphs and exploring related combinatorial sequences.

Contribution

It introduces the functions maxf and minf to quantify facets of symmetric edge polytopes and derives formulas for certain classes of sparse graphs, advancing understanding of their combinatorial properties.

Findings

Formulas for the number of facets in several classes of sparse graphs

Partial progress on conjectures about facet-maximizing graphs

New observations on integer sequences from binomial coefficient sums

Abstract

Symmetric edge polytopes, also called adjacency polytopes, are lattice polytopes determined by simple undirected graphs. We introduce the integer array \(\mathrm{maxf}(n,m)\) giving the maximum number of facets of a symmetric edge polytope for a connected graph having \(n\) vertices and \(m\) edges, and the corresponding sequence \(\mathrm{minf}(n,m)\) of minimal values. We establish formulas for the number of facets obtained in several classes of sparse graphs and provide partial progress toward conjectures that identify facet-maximizing graphs in these classes. These formulas are combinatorial in nature and lead to independently interesting observations and conjectures regarding integer sequences defined by sums of products of binomial coefficients.