The number of independent sets in an irregular graph

TL;DR

This paper proves tight upper and lower bounds on the number of independent sets in irregular graphs, settling a conjecture and extending results to weighted versions, with equality cases characterized.

Contribution

It establishes the first proven bounds on independent sets in irregular graphs, generalizing previous regular graph results and confirming conjectures.

Findings

Proved an upper bound on independent sets in irregular graphs.

Established a lower bound matching the upper bound's conditions.

Extended bounds to weighted independent set polynomials.

Abstract

Settling Kahn's conjecture (2001), we prove the following upper bound on the number $i(G)$ of independent sets in a graph $G$ without isolated vertices: \[ i(G) \le \prod_{uv \in E(G)} i(K_{d_u,d_v})^{1/(d_u d_v)}, \] where $d_u$ is the degree of vertex $u$ in $G$. Equality occurs when $G$ is a disjoint union of complete bipartite graphs. The inequality was previously proved for regular graphs by Kahn and Zhao. We also prove an analogous tight lower bound: \[ i(G) \ge \prod_{v \in V(G)} i(K_{d_v+1})^{1/(d_v + 1)}, \] where equality occurs for $G$ a disjoint union of cliques. More generally, we prove bounds on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph.