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Non-separable graphs meet Ledoux's polynomials

Paul Mansanarez

TL;DR

The paper resolves a conjecture linking the combinatorics of the polynomials \(\operatorname{R}_n\), arising in Ledoux's entropy-derivative framework, to degree sequences of non-separable graphs. By showing that the nonzero coefficient indices \(I_n^*\) coincide with the DNSG(\,n) degree sequences and that DNSG(\,n+1) arises from applying the \(\mathcal{L}\) and \mathcal{H} actions to DNSG(\,n), the authors prove that the number of monomials in \(\operatorname{R}_n\) equals \(d_{ns}(n) - 1\). This result substantiates a bridge between the algebraic structure of multivariate polynomials and combinatorial graph realizations, answering the conjecture of MPS24. The finding deepens understanding of the combinatorics underlying the entropy-derivative polynomials and highlights a new cross-domain connection between information theory-inspired objects and non-separable graphs.

Abstract

In the pathbreaking article \cite{LED16}, an integral representation of the derivatives of entropy along the heat flow of a probability measure was established under suitable moment conditions. These integral representations have found significant applications in diverse domains - notably in information theory (e.g., entropy power inequalities, monotonicity of Fisher information) and in estimation theory (through the link between entropy derivatives and the minimum mean square error, MMSE, in Gaussian channels). The representations involve multivariate polynomials $(R_n)_n$, arising from a Lie algebra framework on multilinear operators. Despite their central role, the combinatorial structure of these polynomials remains only partially understood. In this note, we prove that the number of monomials in $R_n$ coincides with the number of degree sequences with degree sum $2n$ having a non-separable graph realization, thereby resolving a conjecture from \cite{MPS24}, and drawing an interesting link between these two domains.

Non-separable graphs meet Ledoux's polynomials

TL;DR

The paper resolves a conjecture linking the combinatorics of the polynomials , arising in Ledoux's entropy-derivative framework, to degree sequences of non-separable graphs. By showing that the nonzero coefficient indices coincide with the DNSG(\,n) degree sequences and that DNSG(\,n+1) arises from applying the and \mathcal{H} actions to DNSG(\,n), the authors prove that the number of monomials in equals \(d_{ns}(n) - 1\). This result substantiates a bridge between the algebraic structure of multivariate polynomials and combinatorial graph realizations, answering the conjecture of MPS24. The finding deepens understanding of the combinatorics underlying the entropy-derivative polynomials and highlights a new cross-domain connection between information theory-inspired objects and non-separable graphs.

Abstract

In the pathbreaking article \cite{LED16}, an integral representation of the derivatives of entropy along the heat flow of a probability measure was established under suitable moment conditions. These integral representations have found significant applications in diverse domains - notably in information theory (e.g., entropy power inequalities, monotonicity of Fisher information) and in estimation theory (through the link between entropy derivatives and the minimum mean square error, MMSE, in Gaussian channels). The representations involve multivariate polynomials , arising from a Lie algebra framework on multilinear operators. Despite their central role, the combinatorial structure of these polynomials remains only partially understood. In this note, we prove that the number of monomials in coincides with the number of degree sequences with degree sum having a non-separable graph realization, thereby resolving a conjecture from \cite{MPS24}, and drawing an interesting link between these two domains.
Paper Structure (6 sections, 7 theorems, 35 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Degree sequences of non-separable graphs
  4. Structure of the coefficients of $\operatorname{R}_n$
  5. Proof of Proposition \ref{['main']}
  6. Acknowledgments

Key Result

Proposition 1.1

Let $n$ be an integer greater than $2$. Then the number of terms in $\operatorname{R}_n$ is equal to $d_{ns}(n) - 1$.

Figures (2)

  • Figure 1: Non-separable graphs realizations of degree sequences of degree sum $6$
  • Figure 2: Non-separable graphs realizations of degree sequences of degree sum $8$

Theorems & Definitions (13)

  • Proposition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 1
  • Theorem 2
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • ...and 3 more