The number of independent sets in a regular graph

Authors:
Yufei Zhao
Affiliations:
Department of mathematics, massachusetts institute of technology, cambridge, ma 02139, usa (e-mail: yufeiz@mit.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2010

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Abstract

We show that the number of independent sets in an N-vertex, d-regular graph is at most (2d+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.