The number of independent sets in an irregular graph
Author(s)Sah, Ashwin; Sawhney, Mehtaab; Zhao, Yufei
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of combinatorial theory. Series B
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)
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