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Identities of the multi-variate independence polynomials from heaps theory

Deniz Kus, Kartik Singh, R. Venkatesh

TL;DR

This work reframes multi-variate independence polynomials via heaps theory, using the inversion lemma and the Cartier–Foata monoid to derive structural identities. It introduces weight-preserving bijections between word-sets associated with a graph and its stable-path tree, enabling a multi-variate Godsil-type identity that links $I(G-u,\mathbf{x})$ to $I(T_G-u',\mathbf{x})$. A new multi-variate identity involving connected bipartite subgraphs is established, yielding a positive-sum formula and a derivative form that extends Christoffel-Darboux-style identities to the multivariate setting. The results provide a combinatorial, bijective route to fundamental and refined identities for multi-variate independence polynomials with potential implications for real-rootedness and related graph-structure properties.

Abstract

We study and derive identities for the multi-variate independence polynomials from the perspective of heaps theory. Using the inversion formula and the combinatorics of partially commutative algebras we show how the multi-variate version of Godsil type identity as well as the fundamental identity can be obtained from weight preserving bijections. Finally, we obtain a new multi-variate identity involving connected bipartite subgraphs similar to the Christoffel-Darboux type identities obtained by Bencs.

Identities of the multi-variate independence polynomials from heaps theory

TL;DR

This work reframes multi-variate independence polynomials via heaps theory, using the inversion lemma and the Cartier–Foata monoid to derive structural identities. It introduces weight-preserving bijections between word-sets associated with a graph and its stable-path tree, enabling a multi-variate Godsil-type identity that links to . A new multi-variate identity involving connected bipartite subgraphs is established, yielding a positive-sum formula and a derivative form that extends Christoffel-Darboux-style identities to the multivariate setting. The results provide a combinatorial, bijective route to fundamental and refined identities for multi-variate independence polynomials with potential implications for real-rootedness and related graph-structure properties.

Abstract

We study and derive identities for the multi-variate independence polynomials from the perspective of heaps theory. Using the inversion formula and the combinatorics of partially commutative algebras we show how the multi-variate version of Godsil type identity as well as the fundamental identity can be obtained from weight preserving bijections. Finally, we obtain a new multi-variate identity involving connected bipartite subgraphs similar to the Christoffel-Darboux type identities obtained by Bencs.
Paper Structure (4 sections, 4 theorems, 42 equations, 3 figures)

This paper contains 4 sections, 4 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Independence polynomials and word decompositions
  3. Weight preserving bijection and Godsil's identity
  4. Bipartite graphs and positive sum identities

Key Result

Proposition 1

Let $\mathbf{w}\in\mathcal{P}_u(G)$ with $u(\mathbf{w})=1$ and write $N_G(u,\mathbf{w})=\{u_1<\cdots<u_d\}$ where $d=d_G(u, \mathbf{w})$. Then there exist unique $\mathbf{w}_1, \dots, \mathbf{w}_{d}\in \mathcal{P}^{\emptyset}(G)$ such that:

Figures (3)

  • Figure 1: A graph with its stable-path tree.
  • Figure 2: Path graph $P_4$
  • Figure 3: Connected bipartite subgraphs of $P_4$ containing $2$ and $3$

Theorems & Definitions (12)

  • Example
  • Proposition
  • proof
  • Lemma
  • proof
  • Theorem 1
  • proof
  • Proposition
  • proof
  • Remark
  • ...and 2 more