· 2012· Preprint
The number of independent sets in a regular graph
- 104citations
- 2012year
Short summary
The number of independent sets in an N-vertex, d-regular graph is at most (2^(d+1)-1)^(N/2d), a bound sharp for disjoint unions of complete d-regular bipartite graphs. This settles a 1991 conjecture by Alon and a 2001 conjecture by Kahn.
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Key points
- An upper bound of (2^(d+1)-1)^(N/2d) is established for the number of independent sets in an N-vertex, d-regular graph.
- This bound is sharp and achieved by disjoint unions of complete d-regular bipartite graphs.
- The result settles conjectures by Alon (1991) and Kahn (2001).
- The proof offers a short method that reduces the general case to the bipartite case.
AI-generated from the title and abstract; the full text is not read.
Abstract
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. 1.
The authors' abstract, as published at the source., 2012 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics