Degree-sequence bounds for independent sets via multivariate local occupancy

TL;DR

This paper introduces new degree-sequence bounds on the expected size of independent sets in graphs, utilizing spectral analysis and multivariate bounds to improve upon prior methods and extend results to specific graph classes.

Contribution

It develops a novel spectral analysis approach for multivariate local occupancy bounds, bypassing previous limitations and extending bounds to triangle-free graphs and graphs with bounded local maximum average degree.

Findings

Established multivariate lower bounds inspired by conjectures and recent partition function bounds.

Proved univariate bounds for triangle-free graphs extending previous work.

Bounded fugacities by c/Δ, with c as an absolute constant, for the bounds to hold.

Abstract

We present new degree-sequence lower bounds on the expected size of an independent set from the hard-core model. For arbitrary graphs, we establish a multivariate lower bound inspired by a conjecture of the first author and Kang and a recent bound on the multivariate partition function due to Lee and the third author. By applying a novel spectral analysis to the local occupancy linear program, our method successfully bypasses the convergence radius limitations of the cluster expansion and avoids induction. For graphs with bounded local maximum average degree, including triangle-free graphs, we prove a univariate bound extending prior work by a subset of the authors. In both cases our bounds require the fugacities to be upper bounded by $c/\Delta$ where $\Delta$ is the maximum degree of the graph and $c$ is an absolute constant depending on the setting.