The number of independent sets in an irregular graph

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: $$i(G)\le \prod _{uv\in E(G)}i(K_{d_u,d_v}{)}^{1\mathrm{/}(d_u{d}_v)},$$ 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: $$i(G)\ge \prod _{v\in V(G)}i(K_{d_v+1}{)}^{1\mathrm{/}(d_v+1)},$$ 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.