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The number of edges of a symmetric edge polytope

Giulia Codenotti, Roberto Riccardi, Lorenzo Venturello

TL;DR

The paper studies the symmetric edge polytope of a graph and proves a sharp lower bound on its number of edges in terms of basic graph invariants, with a complete characterization of extremal graphs. It then connects these combinatorial bounds to Ehrhart theory, via the h^*-polynomial and its γ-polynomial, and develops a triangulation-based framework to understand how edge deletions affect h^*-polynomials. A key construction is the per-edge analysis that leads to a nonnegative quadratic coefficient z_2(G), tying the edge bound to γ-positivity phenomena and enabling a reduction of the Osugi–Tsuchiya conjecture to 2-connected graphs. The work highlights a rich interaction between graph structure, lattice polytopes, and Ehrhart theory, and opens several avenues for further exploration of γ-positivity and unimodular triangulations in this setting.

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.

The number of edges of a symmetric edge polytope

TL;DR

The paper studies the symmetric edge polytope of a graph and proves a sharp lower bound on its number of edges in terms of basic graph invariants, with a complete characterization of extremal graphs. It then connects these combinatorial bounds to Ehrhart theory, via the h^*-polynomial and its γ-polynomial, and develops a triangulation-based framework to understand how edge deletions affect h^*-polynomials. A key construction is the per-edge analysis that leads to a nonnegative quadratic coefficient z_2(G), tying the edge bound to γ-positivity phenomena and enabling a reduction of the Osugi–Tsuchiya conjecture to 2-connected graphs. The work highlights a rich interaction between graph structure, lattice polytopes, and Ehrhart theory, and opens several avenues for further exploration of γ-positivity and unimodular triangulations in this setting.

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.

Paper Structure

This paper contains 13 sections, 26 theorems, 60 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattice polytopes
  4. Triangulations and relations with Ehrhart theory
  5. Basics of symmetric edge polytopes
  6. Real-rootedness, unimodality and $\gamma$-positivity
  7. Lower bound on the number of edges of symmetric edge polytopes
  8. Characterization of graphs with $z_2(G)=0$
  9. Connection to the Ehrhart theory of symmetric edge polytopes
  10. The $\ell^*$-polynomial of special simplices
  11. A formula for the difference of $h^*$-polynomials when removing an edge
  12. Connection to $z_2(G)$
  13. Further directions

Key Result

Theorem 1

(thm:edges_SEP_ineq) Let $G=(V,E)$ be a connected graph. Then with $E_3(G):= \lbrace e \in E \, | \, e \text{ belongs to a 3-cycle of $G$} \rbrace$.

Figures (11)

  • Figure 1: $Z$ decreases: $Z(G_i,l)=2$, meanwhile $Z(G_{i+1},l)=0$
  • Figure 2: $Z(G_i,l)= 0 = Z(G_{i+1},l)$
  • Figure 3: $Z$ increases: $Z(G_i,l)=2$, meanwhile $Z(G_{i+1},l)=6$
  • Figure 8: An example of $P \sim Q$, where $P= \lbrace l, g, h, f \rbrace$ and $Q = \lbrace l,g, r, s \rbrace$
  • Figure 9: The two possible subgraphs contained in $G$
  • ...and 6 more figures

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Conjecture 1.1
  • Definition 1.2
  • Conjecture 3
  • Theorem 4
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 43 more