The number of independent sets in an irregular graph
Name
1805.04021.pdf
Description
Accepted version
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573.62 KB
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Checksum (MD5)
fe687e864ce449355637cba091e205bb
Author(s)
Sah, Ashwin
Sawhney, Mehtaab
Zhao, Yufei
Date Issued
September 2019
Journal
Journal of combinatorial theory. Series B
Publisher
Elsevier BV
Citation
Sah, Ashwin et al. “The number of independent sets in an irregular graph.” Journal of combinatorial theory. Series B, vol. 138, 2019, pp. 172-195 © 2019 The Author(s)
Version
Author's final manuscript
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)≤∏uv∈E(G)i(Kdu,dv)1/(dudv), where du 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(Kdv+1)1/(dv+1), where equality occurs for G a disjoint union of cliques. More generally, we prove bounds on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/J.JCTB.2019.01.007
