Matchings and independent sets of a fixed size in regular graphs

Carroll, Teena; Galvin, David; Tetali, Prasad

doi:10.1016/j.jcta.2008.12.008

Cited by 18 publications

(29 citation statements)

References 13 publications

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“…For α ≤ (d + 1) −2 , Corollary 3 improves the bound given by Carroll, Galvin, and Tetali in [3] 2 . Specializing to Z d (n) and Q d we get upper bounds of α (1 − log α − α(2d + 1)/2) and α (1 − log α − α(d + 1)/2) respectively on the normalized logarithm of the number of independent sets of size αn.…”

Section: The Hard-core Modelsupporting

confidence: 67%

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Birthday inequalities, repulsion, and hard spheres

Perkins

2016

Proc. Amer. Math. Soc.

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We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does not hold in general. We give three different applications of the birthday inequality in statistical physics and combinatorics: we prove lower bounds on the free energy of the hard sphere model and upper bounds on the number of independent sets and matchings of a given size in d-regular graphs.The birthday inequality is implied by a repulsion inequality: the expected volume of the union of spheres of radius r around n randomly placed centers increases if we condition on the event that the centers are at pairwise distance greater than r. Surprisingly we show that the repulsion inequality is not true in general, and in particular that it fails in 24-dimensional Euclidean space: conditioning on the pairwise repulsion of centers of 24-dimensional spheres can decrease the expected volume of their union.

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Section: The Hard-core Modelsupporting

confidence: 67%

“…For this range of α, the best upper bound in [3] on IS(αn) is the third bound given in Theorem 1.6, 2 αn n/2 αn . On the scale of the free energy this is −α log(α) − (1/2 − α) log(1 − 2α).…”

Section: The Hard-core Modelmentioning

confidence: 96%

Birthday inequalities, repulsion, and hard spheres

Perkins

2016

Proc. Amer. Math. Soc.

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“…Using this Heilmann-Lieb inequality, we shall derive rigorous upper bounds for N (i) of general graphs. In the case of regular graphs they are stricter than those in [17] in a region with small dimer density; in the case of general graphs, we obtain an upper bound for the matching generating polynomial which is stricter than the one found in [18].…”

Section: Introductionmentioning

confidence: 69%

“…We have shown that this inequality leads to rigorous upper bounds for the number of matchings N (i) improving those known up to now for regular graphs [32] in the region of low dimer density, and for general graphs. In the latter case, we derive an upper bound for the matching matching polynomial improving the one in [18].…”

Section: Discussionmentioning

confidence: 99%

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

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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.

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“…The above proof is essentially the same as the proofs in Carroll, Galvin, and Tetali with the small observation that

λ

can be chosen so that

k

is the most likely size of a matching (or independent set) drawn from

H_{d, n}

. The factor

2 \sqrt{n}

in both cases can surely be improved by using some regularity of the independent set and matchings sequence of a general

d

‐regular graph; we leave this for future work.…”

Section: Independent Sets and Matchings Of A Given Sizementioning

confidence: 90%

Independent sets, matchings, and occupancy fractions

Davies

Jenssen

Perkins

et al. 2017

Journal of London Math Soc

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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.

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Matchings and independent sets of a fixed size in regular graphs

Cited by 18 publications

References 13 publications

Birthday inequalities, repulsion, and hard spheres

Birthday inequalities, repulsion, and hard spheres

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

Independent sets, matchings, and occupancy fractions

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