On the Validations of the Asymptotic Matching Conjectures |
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| Authors: | S. Friedland E. Krop P. H. Lundow K. Markström |
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| Affiliation: | 1.Department of Mathematics, Statistics and Computer Science,University of Illinois at Chicago,Chicago,USA;2.Berlin Mathematical School,Berlin,Germany;3.Department of Physics,AlbaNova University Center,Stockholm,Sweden;4.Department of Mathematics and Mathematical Statistics,Ume? University,Ume?,Sweden |
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| Abstract: | In this paper we review the asymptotic matching conjectures for r-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density p, for any p∈[0,1], for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite r-regular bipartite graphs. |
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| Keywords: | Matching and asymptotic growth of average matchings for r-regular bipartite graphs Monomer-dimer partitions and entropies |
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