Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graph minors, Ehrhart theory, and a monotonicity property

Tamás Kálmán, Lilla Tóthmérész

TL;DR

This paper analyzes extended root polytopes $\tilde{\mathcal{Q}}_D$ associated to directed graphs, focusing on their $h^*$-polynomials within Ehrhart theory. It derives a formula expressing $h^*_D$ as a generating function over dissecting forests weighted by a passivity statistic, and proves coefficientwise monotonicity of $h^*$ under edge deletion and contraction using dissections built from acyclic circuit signatures. The authors also establish multiplicativity over disjoint unions and characterize equality cases, including a rich Gorenstein theory: contracting a minimal dijoin preserves the $h^*$-polynomial and leads to reflexive quotient polytopes, with special simplices providing a geometric bridge to BN mirror symmetry. Overall, the work connects graph minors, dissections, and Gorenstein properties to deepen understanding of the Ehrhart theory of root polytopes and their combinatorial invariants.

Abstract

We study the extended root polytope associated to a directed graph. We show that under the operations of deletion and contraction of an edge of the graph, none of the coefficients of the $h^*$-polynomial of the associated extended root polytope increase. We examine cases when the $h^*$-polynomial does not change, for instance when contracting the edges of a minimal directed join in a digraph whose lattice polytope has the Gorenstein property.

Graph minors, Ehrhart theory, and a monotonicity property

TL;DR

This paper analyzes extended root polytopes associated to directed graphs, focusing on their -polynomials within Ehrhart theory. It derives a formula expressing as a generating function over dissecting forests weighted by a passivity statistic, and proves coefficientwise monotonicity of under edge deletion and contraction using dissections built from acyclic circuit signatures. The authors also establish multiplicativity over disjoint unions and characterize equality cases, including a rich Gorenstein theory: contracting a minimal dijoin preserves the -polynomial and leads to reflexive quotient polytopes, with special simplices providing a geometric bridge to BN mirror symmetry. Overall, the work connects graph minors, dissections, and Gorenstein properties to deepen understanding of the Ehrhart theory of root polytopes and their combinatorial invariants.

Abstract

We study the extended root polytope associated to a directed graph. We show that under the operations of deletion and contraction of an edge of the graph, none of the coefficients of the -polynomial of the associated extended root polytope increase. We examine cases when the -polynomial does not change, for instance when contracting the edges of a minimal directed join in a digraph whose lattice polytope has the Gorenstein property.
Paper Structure (13 sections, 22 theorems, 25 equations, 2 figures)

This paper contains 13 sections, 22 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph notations
  4. Definition of the $h^*$-polynomial
  5. The extended root polytope
  6. A formula for the $h^*$-polynomial
  7. Circuit signatures and dissections
  8. Acyclic circuit signatures
  9. Dissections via acyclic circuit signatures
  10. Proof of Theorem \ref{['thm:h_of_contraction']} and a product formula
  11. The case of equality
  12. Gorenstein extended root polytopes
  13. Proof of Theorem \ref{['thm:interior_poly_fixed_order']}

Key Result

Theorem 1.1

Let $D$ be a directed graph. For any edge $e$ of $D$, we have that $h^*_{{D-e}}$ is coefficientwise smaller than or equal to $h^*_{D}$.

Figures (2)

  • Figure 1: Illustration for Example \ref{['ex']}.
  • Figure 2: Illustration for Example \ref{['ex:feloldasok']}.

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Stanley's monotonicity theorem, Stanley_monotonicity
  • proof : Proof of Theorem \ref{['thm:h_of_deletion']}
  • Proposition 1.4
  • Proposition 1.5
  • Claim 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • ...and 48 more