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 results of Kahn @cite_27 @cite_1 , Galvin and Tet ali @cite_19 , and Zhao @cite_7 culminating in the fact that @math is maximized over @math -regular graphs by @math are based on the entropy method, a powerful tool for the type of problems we address here. Apart from the results mentioned above, see @cite_10 and @cite_15 for surveys of the method. A direct application of the method requires the graph @math to be bipartite. Zhao @cite_25 showed that in some, but not all cases, this restriction can be removed by using a bipartite swapping trick'. An entropy-free proof of Galvin and Tet ali's general theorem on counting homomorphisms was recently given by Lubetzky and Zhao @cite_26 . Our method also does not use entropy, but in contrast to the other proofs it works directly for all @math -regular graphs, without a reduction to the bipartite case. The method deals directly with the hard-core model instead of counting homomorphisms and seems to require more problem-specific information than the entropy method; a question for future work is to extend the method to a more general class of homomorphisms.
@cite_26 @cite_7 @cite_1 @cite_19 @cite_27 @cite_15 @cite_10 @cite_25
2963702702 2144865572 2083618772 2409645877
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 Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. 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 Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � 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. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees. We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size @? in a d-regular graph on N vertices. For [email protected]?N bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size @? in the graph consisting of N2d disjoint copies of K"d","d. This provides asymptotic evidence for a conjecture of S. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets. We show that the Graph Isomorphism (GI) problem and the more general problems of String Isomorphism (SI) andCoset Intersection (CI) can be solved in quasipolynomial(exp((logn)O(1))) time. The best previous bound for GI was exp(O( √n log n)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp(O (√ n)), where n is the size of the permutation domain (Babai, 1983). Following the approach of Luks’s seminal 1980 82 paper, the problem we actually address is SI. This problem takes two strings of length n and a permutation group G of degree n (the “ambient group”) as input (G is given by a list of generators) and asks whether or not one of the strings can be transformed into the other by some element of G. Luks’s divide-and-conquer algorithm for SI proceeds by recursion on the ambient group. We build on Luks’s framework and attack the obstructions to efficient Luks recurrence via an interplay between local and global symmetry. We construct group theoretic “local certificates” to certify the presence or absence of local symmetry, aggregate the negative certificates to canonical k-ary relations where k = O(log n), and employ combinatorial canonical partitioning techniques to split the k-ary relational structure for efficient divide-and- conquer. We show that in a well–defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. The central element of the algorithm is the “local certificates” routine which is based on a new group theoretic result, the “Unaffected stabilizers lemma,” that allows us to construct global automorphisms out of local information.