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Independent sets, matchings, and occupancy fractions

Davies, Ewan, Jenssen, Matthew, Perkins, Will and Roberts, Barnaby (2017) Independent sets, matchings, and occupancy fractions. Journal of the London Mathematical Society, 96 (1). pp. 47-66. ISSN 0024-6107

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Identification Number: 10.1112/jlms.12056


We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tetali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree

Item Type: Article
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Additional Information: © 2017 London Mathematical Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 02 Feb 2018 16:04
Last Modified: 20 Oct 2021 00:55
Projects: EP/P009913/1
Funders: Engineering and Physical Sciences Research Council

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