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arXiv · 0909.3354

The Number of Independent Sets in a Regular Graph

Yufei Zhao
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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.

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Yufei Zhao. 2009-09-18. The Number of Independent Sets in a Regular Graph. https://doi.org/10.1017/s0963548309990538

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