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Abstract  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 the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of    K   d  ,  dmaximizes the number of independent sets and the independence polynomial of a   d  ‐regular graph.  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    K   d  ,  d    . Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström.  In probabilistic language, our main theorems state that for all   d  ‐regular graphs and all   λ  , the occupancy fraction of the hard‐core model and the edge occupancy fraction of the monomer‐dimer model with fugacity   λ   are maximized by    K   d  ,  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.

Independent sets, matchings, and occupancy fractions