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.

Abstract. We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on n vertices with maximum degree d, showing that an independent set drawn uniformly at random from such a graph has expected size at leastn. This gives an alternative proof of Shearer's upper bound on the Ramsey number R(3, k). We then prove that the total number of independent sets in a triangle-free graph with maximum degree d is at least expn . The constant 1/2 in the exponent is best possible. In both cases, tightness is exhibited by a random d-regular graph.Both results come from considering the hard-core model from statistical physics: a random independent set I drawn from a graph with probability proportional to λ |I| , for a fugacity parameter λ > 0. We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree d. The bound is asymptotically tight in d for all λ = O d (1).We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory.

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti‐ferromagnetic Potts model on cubic graphs at every temperature and for all q≥2. This immediately implies corresponding tight bounds on the anti‐ferromagnetic Potts partition function. Taking the zero‐temperature limit gives new results in extremal combinatorics: the number of q‐colorings of a 3‐regular graph, for any q≥2, is maximized by a union of K3,3's. This proves the d = 3 case of a conjecture of Galvin and Tetali.

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

For a graph G = (V, E), k ∈ N, and a complex number w the partition function of the univariate Potts model is defined aswhere [k] := {1, . . . , k}. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any ∆ ∈ N and any k ≥ e∆ + 1, there exists an open set U in the complex plane that contains the interval [0, 1) such that Z(G; k, w) = 0 for any w ∈ U and any graph G of maximum degree at most ∆. (Here e denotes the base of the natural logarithm.) For small values of ∆ we are able to give better results.As an application of our results we obtain improved bounds on k for the existence of deterministic approximation algorithms for counting the number of proper k-colourings of graphs of small maximum degree.