# The number of independent sets in an irregular graph

@article{Sah2019TheNO, title={The number of independent sets in an irregular graph}, author={Ashwin Sah and Mehtaab Sawhney and David Stoner and Yufei Zhao}, journal={J. Comb. Theory, Ser. B}, year={2019}, volume={138}, pages={172-195} }

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 ) ≤ ∏ u v ∈ 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 ) ≥ ∏ v ∈ V ( G ) i ( K d v + 1… CONTINUE READING

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