Number of Independent Sets in Regular and Irregular Graphs: A 31 Year Journey

TL;DR

This paper reviews 31 years of progress in bounding the number of independent sets in regular and irregular graphs, highlighting key contributions and their incremental improvements.

Contribution

It synthesizes and clarifies the main results from various researchers, emphasizing the evolution of upper bounds for independent sets in graphs.

Findings

Progressively stronger upper bounds established

Main results reproduced with clearer explanations

Focus on unweighted case ($mbda=1$) for intuition

Abstract

We review the progress made on bounding the number of independent sets in $d$-regular and irregular graphs over the last 31 years. We particularly focus on contributions from Kahn, Zhao, and Sah et al. in incrementally proving stronger and more general versions of the upper bound. We reproduce the main results of these works, particularly focusing on the unweighted special case (with fugacity $\lambda = 1$), which allows us to provide more intuitive and clear explanations of the key ideas that have been developed in the field over three decades.