Independent Sets, Matchings, and Occupancy Fractions

TL;DR

This paper establishes tight upper bounds on the independence and matching polynomials of d-regular graphs, showing that unions of complete bipartite graphs maximize these polynomials and related occupancy fractions, confirming conjectures and strengthening prior results.

Contribution

It provides new tight bounds on independence and matching polynomials for d-regular graphs, proving the asymptotic upper matching conjecture and extending previous work.

Findings

Unions of K_{d,d} maximize independence and matching polynomials.

The occupancy fractions in the hard-core and monomer-dimer models are maximized by K_{d,d}.

Confirmed the asymptotic upper matching conjecture.

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 $K_{d,d}$ maximizes 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 Markstr\"om. In probabilistic language, our main theorems state that for all d-regular graphs and all $\lambda$, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity $\lambda$ 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.