On the Validations of the Asymptotic Matching Conjectures

Friedland, Shmuel; Krop, Elliot; Lundow, Per-Håkan; Markström, Klas

doi:10.1007/s10955-008-9550-y

Cited by 26 publications

(53 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1,2) are satisfied by the approximate distribution of bipartite regular random-graph obtained in [19]. In the case of bipartite regular biconnected graphs, we argue that a conjecture on the entropy for these graphs, made in [19,25], implies the conjecture that for v → ∞ the frequency of the violations tends to zero.…”

Section: Introductionmentioning

confidence: 77%

Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

Butera

Federbush

Pernici

2015

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

Using a simple relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer-dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjecture that the virial expansion coefficients m k are always positive.These considerations can be extended to the study of related bounds on finite graphs generalizing the infinite regular lattices, namely the finite grids and the regular biconnected graphs. The validity of the bounds ∆ k ln(i!N (i)) ≤ 0 for k ≥ 2, where N (i) is the number of configurations of i dimers on the graph and ∆ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients.Our tests on many finite lattice graphs indicate that on large lattices these bounds are satisfied, giving support to the conjecture on the positivity of the virial coefficients. Moreover, in an exhaustive survey of some classes of regular biconnected graphs with a not too large number v of vertices, we observe only few violations of these bounds. We conjecture that the frequency of the violations vanishes as v → ∞.Using an inequality by Heilmann and Lieb, we find rigorous upper bounds on N (i) valid for arbitrary graphs and for regular graphs. The similarity between this inequality and the one conjectured above suggests that one study the stricter inequality m k ≥ 1 2k for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models.

show abstract

Section: Introductionmentioning

confidence: 77%

Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

Butera

Federbush

Pernici

2015

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

show abstract

“…In [7] an asymptotic variant of Conjecture 1.2 is presented. Let {G k } be a sequence of d-regular bipartite graphs with |V k |, the number of vertices of G k , growing to infinity, and fix α ∈ [0, 1].…”

Section: Introductionmentioning

confidence: 99%

Matchings and independent sets of a fixed size in regular graphs

Carroll

Galvin

Tetali

2009

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

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 2 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 N 2d disjoint copies of K d,d . This provides asymptotic evidence for a conjecture of S. Friedland et al. 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.

show abstract

“…The middle term is obtained by considering the eigenvalues of A 3,5 , the 10 by 10 matrix whose row and columns are indexed by lexicographically ordered 3-element subsets of [5]. (See [9][10][11].) From the previous section we deduce h(m) = log ρ(m) 2m .…”

Section: The Transfer Matrix Methods For Counting Perfect Matchings Inmentioning

confidence: 99%

Matchings in m-generalized fullerene graphs

2015

View full text Add to dashboard Cite

A connected planar graph is called m-generalized fullerene if two of its faces are mgons and all other faces are pentagons and hexagons. In this paper we first determine some structural properties of m-generalized fullerenes and then use them to obtain new results on the enumerative aspects of perfect matchings in such graphs. We provide both upper and lower bounds on the number of perfect matchings in m-generalized fullerene graphs and state exact results in some special cases.

show abstract

On the Validations of the Asymptotic Matching Conjectures

Cited by 26 publications

References 31 publications

Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

Matchings and independent sets of a fixed size in regular graphs

Matchings in m-generalized fullerene graphs

Contact Info

Product

Resources

About