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 Tet ali, and Zhao showing that a union of copies of @math 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 @math . 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 @math , the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity @math are maximized by @math . 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.
The following question is due to Chatterjee and Varadhan 2011. Fix 0<p<r<1 and take G Gn,p, the Erdi¾?s-Renyi random graph with edge density p, conditioned to have at least as many triangles as the typical Gn,r. Is G close in cut-distance to a typical Gn,r? Via a beautiful new framework for large deviation principles in Gn,p, Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of p,r where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d-regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all p,r such that rd,hpr lies on the convex minorant of xi¾?hpx1 d where hp is the rate function of a binomial with parameter p. In particular, the answer for triangles involves hpx rather than the natural guess of hpx1 3 where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of G Gn,p is conditioned to exceed the typical value of the largest eigenvalue of Gn,r. Building on the work of Chatterjee and Diaconis 2012 we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn 2001 and Galvin and Tet ali 2004. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 109-146, 2015 We show that the number of independent sets in an N-vertex, d-regular graph is at most (2d+1 − 1)N 2d, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also works for a weighted generalization, i.e., an upper bound for the independence polynomial of a regular graph. For n-regular, N-vertex bipartite graphs with bipartition A U B, a precise bound is given for the sum over independent sets I of the quantity μ |I ∩ A| λ |I ∩ B| , (In other language, this is bounding the partition function for certain instances of the hard-core model.) This result is then extended to graded partially ordered sets, which in particular provides a simple proof of a well-known bound for Dedekind's Problem given by Kleitman and Markowsky in 1975. For given graphs G and H, let |Hom(G,H)| denote the set of graph homomorphisms from G to H. We show that for any finite, n-regular, bipartite graph G and any finite graph H (perhaps with loops), |Hom(G,H)| is maximum when G is a disjoint union of Kn,n’s. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the logarithm of |Hom(G,H)| in terms of a simply expressed parameter of H. We also consider weighted versions of these results which may be viewed as statements about the partition functions of certain models of physical systems with hard constraints. We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ∣I∣ for some fixed λ > 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices. We explain the notion of the entropy of a discrete random variable, and derive some of its basic properties. We then show through examples how entropy can be useful as a combinatorial enumeration tool. We end with a few open questions.  We provide an upper bound to the number of graph homomorphisms from G to H, where H is a fixed graph with certain properties, and G varies over all N-vertex, d-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tet ali, who studied the number of graph homomorphisms from G to H when G is bipartite. We also apply our techniques to graph colorings and stable set polytopes.