Independent sets, matchings, and occupancy fractions

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 Kd,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 Kd,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 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.

Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

“…It is conjectured that latter maximizes the number of independents of every fixed size. Let i t (G) denote the number of independent sets of size t in G. Recall that kG denotes a disjoint union of k copies of G. See [23,Section 8] for the current best bounds on this problem. 9.2.…”

“…The proof in Section 3, following [45], used a variant of the Hölder's inequality, and is related to the original entropy method proof. Recently, an elegant new proof of the result was found [23] using a novel method, unrelated to previous proofs. We discuss this new technique in this section.…”

“…Notably, Davies, Jenssen, Perkins, and Roberts [23] recently gave a new proof of the above theorems by introducing a powerful new technique, which has already had a number Homomorphisms from G to correspond to independent sets of G.…”

“…In [23], Theorem 7.2 was proved for all d-regular graphs G by considering all graphs on d vertices that could be induced by the neighborhood of a vertex in G and using a linear program to constrain the probability distribution of the neighborhood profile of a random vertex. When G is trianglefree, the neighborhood of a vertex is always an independent set, which significantly simplifies the situation.…”

“…They are both based on the following idea, introduced in [23] for this problem. We draw a random independent set I from the hard-core model and look at the neighborhood of a uniform random vertex v ∈ V (G).…”