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Log-Concavity of the Grothendieck Classes of Banana Graphs and Clasped Necklaces

Stephanie Chen

TL;DR

The paper proves log-concavity for the Grothendieck classes of banana graphs and for the four polynomials used in the melonic recursive formula, showing that these classes are polynomials in $\mathbb{S}=[\mathcal{M}_{0,4}]$ with nonnegative coefficients and LC. It further analyzes ultra-log-concavity and how these properties interact under products and shifts, with detailed parity-dependent results and computational evidence up to $m=10$. Extending the framework to necklace graphs, the authors derive closed forms for clasped necklaces and prove their LC in $\mathbb{S}$, strengthening the log-concavity conjecture in this melonic setting. The work combines deletion-contraction recurrences, explicit polynomial manipulations, and coefficient analyses (supported by Sage computations) to illuminate the combinatorial-algebraic structure of graph hypersurface complements in the Grothendieck ring and their log-concavity properties. Overall, the results provide a concrete verification of log-concavity for a broad family of melonic graphs and pave the way for further exploration of ultra-log-concavity in this motivic context, with potential implications for algebro-geometric Feynman rules and moduli-space connections.

Abstract

The Grothendieck classes of melonic graphs satisfy a recursive relation and may be written as polynomials in the class of the moduli space $\mathcal{M}_{0,4}$ with nonnegative integer coefficients, conjectured to be log-concave. In this article, we investigate log-concavity and ultra-log-concavity for the Grothendieck class of banana graphs and the three families of polynomials involved in the recursive relation. We prove that all four are log-concave, establishing the specific case of banana graphs for the log-concavity conjecture. We additionally introduce the infinite family of clasped necklaces, melonic graphs obtained by replacing an edge of a $2$-banana with a string of $m$-bananas. Using the recursive relation, we explicitly compute the classes of clasped necklaces and prove that they too are log-concave.

Log-Concavity of the Grothendieck Classes of Banana Graphs and Clasped Necklaces

TL;DR

The paper proves log-concavity for the Grothendieck classes of banana graphs and for the four polynomials used in the melonic recursive formula, showing that these classes are polynomials in with nonnegative coefficients and LC. It further analyzes ultra-log-concavity and how these properties interact under products and shifts, with detailed parity-dependent results and computational evidence up to . Extending the framework to necklace graphs, the authors derive closed forms for clasped necklaces and prove their LC in , strengthening the log-concavity conjecture in this melonic setting. The work combines deletion-contraction recurrences, explicit polynomial manipulations, and coefficient analyses (supported by Sage computations) to illuminate the combinatorial-algebraic structure of graph hypersurface complements in the Grothendieck ring and their log-concavity properties. Overall, the results provide a concrete verification of log-concavity for a broad family of melonic graphs and pave the way for further exploration of ultra-log-concavity in this motivic context, with potential implications for algebro-geometric Feynman rules and moduli-space connections.

Abstract

The Grothendieck classes of melonic graphs satisfy a recursive relation and may be written as polynomials in the class of the moduli space with nonnegative integer coefficients, conjectured to be log-concave. In this article, we investigate log-concavity and ultra-log-concavity for the Grothendieck class of banana graphs and the three families of polynomials involved in the recursive relation. We prove that all four are log-concave, establishing the specific case of banana graphs for the log-concavity conjecture. We additionally introduce the infinite family of clasped necklaces, melonic graphs obtained by replacing an edge of a -banana with a string of -bananas. Using the recursive relation, we explicitly compute the classes of clasped necklaces and prove that they too are log-concave.

Paper Structure

This paper contains 20 sections, 16 theorems, 136 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Banana graphs
  3. Log-concavity
  4. Summary of paper
  5. Grothendieck classes of melonic graphs
  6. Grothendieck ring of varieties
  7. Grothendieck class of a graph
  8. Melonic graphs
  9. Log-concavity and related properties
  10. Recursive Polynomials
  11. Log-concavity of the recursion polynomials
  12. Ultra log-concavity for the recursion polynomials
  13. Ultra-log-concavity of order $m$ for the recursion polynomials
  14. Necklace Graphs
  15. Necklace graph of $2$-bananas
  16. ...and 5 more sections

Key Result

Theorem 1.1

For all $n\geq 0$, the Grothendieck classes are all log-concave as polynomials in $\mathbb{S}$.

Figures (2)

  • Figure 1: The clasped necklace $G'_{3,7}$, obtained by replacing an edge of a $2$-banana with a string of $6$$3$-bananas.
  • Figure 2: The clasped necklace $G'_{3,7}$ with $6$$3$-bananas.

Theorems & Definitions (34)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1
  • Example 1: Banana graphs
  • Definition 2.2
  • Conjecture 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • ...and 24 more