Christoffel–Darboux Type Identities for the Independence Polynomial

Bencs, Ferenc

doi:10.1017/s0963548318000135

Cited by 7 publications

(14 citation statements)

References 7 publications

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“…The first part of the lemma follows by induction on

r

from the fact that

P false[u_{1}, u_{2} \in I false] > P false[u_{1} \in I false] \cdot P false[u_{2} \in I false]

when

u_{1}, u_{2}

are in the same connected component and in the same part of the bipartition of

G

. In this is shown to be a consequence of the FKG inequality; see also and Corollary 1.5 of . An intuitive reason for this fact (which can be turned into a rigorous argument using Weitz's tree ), is that conditioning on the event that a vertex

v

is occupied forbids its neighbors from being in the independent set; conditioning on the event that

v

is not occupied increases the probability each of its neighbors are occupied, and these effects propagate through the bipartite graph.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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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“…The first part of the lemma follows by induction on

r

from the fact that

P false[u_{1}, u_{2} \in I false] > P false[u_{1} \in I false] \cdot P false[u_{2} \in I false]

when

u_{1}, u_{2}

are in the same connected component and in the same part of the bipartition of

G

v

is occupied forbids its neighbors from being in the independent set; conditioning on the event that

v

is not occupied increases the probability each of its neighbors are occupied, and these effects propagate through the bipartite graph.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Independent sets, matchings, and occupancy fractions

Davies

Jenssen

Perkins

et al. 2017

Journal of London Math Soc

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show abstract

“…Note that G × K 2 is also a d-regular graph. Applying the inequality of Statement holding for bipartite graphs [96], by (18) we have for all x ≤ 0…”

Section: Applications In Graph Theorymentioning

confidence: 99%

“…This result was proved by M. Chudnovsky and P. Seymour in 2004 (published in 2007). After that, a lot of other proofs of Theorem 11.1 appear [18,128,133].…”

Section: Real-rootedness Of P C-polynomialmentioning

confidence: 99%

“…Let us state them in more general case. Lemma 12.1 [18]. Let G be a graph, v ∈ V (G), B v be the set of induced connected, bipartite subgraphs of G containing the vertex v.…”

Section: Chudnovsky Seymour Theoremmentioning

confidence: 99%

“…If G is a claw-free graph then all roots of I(G, x) are real. We follow the proof of F. Bencs [18] (maybe, the simplest one).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

$PC$-polynomial of graph

Gubarev

2018

Preprint

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We define P C-polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root β(G) of P C-polynomial of the corresponding graph.The random algebra is defined in such way that its growth rate equals the largest root of P C-polynomial of random graph. We prove that for almost all graphs all sufficiently large real roots of P C-polynomial lie in neighbourhoods of roots of P Cpolynomial of random graph. We show how to calculate the series expansions of the latter roots. The average value of β(G) over all graphs with the same number of vertices is computed.We found the graphs on which the maximal value of β(G) with fixed numbers of vertices and edges is reached. From this, we derive the upper bound of β(G). Modulo one assumption, we do the same for minimal value of β(G).We study the Nordhaus Gaddum bounds of β(G) + β( Ḡ) and β(G)β( Ḡ).

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Absence of zeros implies strong spatial mixing

Regts

2023

Probab. Theory Relat. Fields

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In this paper we show that absence of complex zeros of the partition function of the hard-core model on any family of bounded degree graphs that is closed under taking induced subgraphs implies that the associated probability measure, the hard-core measure, satisfies strong spatial mixing on that family. As a corollary we obtain that the hard-core measure on the family of bounded degree claw-free graphs satisfies strong spatial mixing for every value of the fugacity parameter. We furthermore derive strong spatial mixing for graph homomorphism measures from absence of zeros of the graph homomorphism partition function.

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Christoffel–Darboux Type Identities for the Independence Polynomial

Cited by 7 publications

References 7 publications

Independent sets, matchings, and occupancy fractions

Independent sets, matchings, and occupancy fractions

$PC$-polynomial of graph

Absence of zeros implies strong spatial mixing

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