The Number of Independent Sets in a Regular Graph

TL;DR

This paper proves a sharp upper bound on the number of independent sets in regular graphs, settling a longstanding conjecture and providing a simplified proof that extends to weighted cases.

Contribution

It establishes a tight bound for all regular graphs, generalizes previous bipartite results, and introduces a simplified proof approach.

Findings

Proves the maximum number of independent sets in d-regular graphs is (2^{d+1} - 1)^{N/2d}.

Extends the bound to weighted independence polynomials.

Provides a short proof reducing the general case to bipartite graphs.

Abstract

We show that the number of independent sets in an N-vertex, d-regular graph is at most (2^{d+1} - 1)^{N/2d}, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also works for a weighted generalization, i.e., an upper bound for the independence polynomial of a regular graph.