Lower bounds for multivariate independence polynomials and their generalisations
Joonkyung Lee, Jaehyeon Seo
TL;DR
This work establishes general lower bounds for the multivariate independence polynomial $Z_G(m{\lambda})$ of a graph $G$, showing $Z_G(\bm{\lambda}) \ge \prod_{v} Z_{K_{d_v+1}}(\lambda_v)^{1/(d_v+1)}$, with $d_v$ the degree of $v$. It extends to a two-spin semiproper colouring partition function $Z_G^{(2)}(\bm{\lambda},\bm{\mu})$, proving $Z_G^{(2)}(\bm{\lambda},\bm{\mu}) \ge \prod_{v} Z^{(2)}_{K_{d_v+1}}(\lambda_v,\mu_v)^{1/(d_v+1)}$ by developing a convex-dual framework based on the dual set $\mathcal{S}_d$ and the concavity of $A_{d+1}(x,y)^{1/(d+1)}$. The paper then advances towards Conjecture 1 by proving the bound for graphs with maximum degree $\Delta(G)\le 2$ using spectral decompositions of antiferromagnetic target graphs $H$, thereby handling paths and cycles. In addition to these main results, the authors discuss extensions to more colours $q$, negative fugacity, and occupancy-fraction perspectives, and reflect on AI-assisted reasoning employed in parts of the proofs. Overall, the work both strengthens bounds for multivariate partition functions and maps a path toward broader antiferromagnetic counting inequalities in graph theory.
Abstract
In statistical physics, the multivariate hard-core model describes a system of particles, each of which receives its own fugacity. In graph-theoretic language, the partition function of the model translates to the multivariate independence polynomial, i.e., the multiaffine generalisation of the independence polynomial, defined by $Z_G(λ_1,\dots,λ_n) := \sum_{I\in\mathcal{I}(G)} \prod_{v\in I}λ_v$, where $\mathcal{I}(G)$ denotes the set of all independent sets in a graph $G$ on $[n]:=\{1,2,\dots,n\}$. We prove that for every simple graph $G$ on $[n]$ and $λ_1,\dots,λ_n\geq 0$, \[ Z_G(λ_1,\dots,λ_n) \geq \prod_{i=1}^n (1+(d_i+1)λ_i)^{1/(d_i+1)}, \] where $d_1,\dots,d_n$ is the degree sequence of $G$. This generalises a result of Sah, Sawhney, Stoner, and Zhao, who proved the univariate case $λ_1=\dots=λ_n=λ$. We further conjecture that our inequality should generalise to other antiferromagnetic models and give some evidence in support of it. In particular, for $λ_i,μ_i\geq 0$, $1\leq i\leq n$, we obtain a stronger inequality \[ \sum_{\substack{I,J\in \mathcal{I}(G) \\ I\cap J=\emptyset}} \prod_{v\in I}λ_v\prod_{u\in J}μ_u \geq \prod_{i=1}^n \left(1+(d_i+1)(λ_i+μ_i)+d_i(d_i+1)λ_iμ_i\right)^{1/(d_i+1)}, \] which proves our conjecture for a multiaffine generalisation of the semiproper colouring partition function with two proper colours. Our key technical steps for both theorems are obtained by using a custom mathematical research agent built on top of Gemini Deep Think, which can be seen as a benchmark demonstrating that the current state-of-the-art language models can, in part, assist with mathematical research.