2019
DOI: 10.1307/mmj/1541667626
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On a Conjecture of Sokal Concerning Roots of the Independence Polynomial

Abstract: A conjecture of Sokal [23] regarding the domain of non-vanishing for independence polynomials of graphs, states that given any natural number ∆ ≥ 3, there exists a neighborhood in C of the interval [0, (∆−1) ∆−1 (∆−2) ∆ ) on which the independence polynomial of any graph with maximum degree at most ∆ does not vanish. We show here that Sokal's Conjecture holds, as well as a multivariate version, and prove optimality for the domain of non-vanishing. An important step is to translate the setting to the language o… Show more

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Cited by 78 publications
(128 citation statements)
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“…We remark that a similar approach was used by the authors in [24] to answer a question of Sokal concerning the location of zeros of the independence polynomial, a.k.a., the partition function of the hard-core model.…”
Section: Our Main Results Arementioning
confidence: 95%
“…We remark that a similar approach was used by the authors in [24] to answer a question of Sokal concerning the location of zeros of the independence polynomial, a.k.a., the partition function of the hard-core model.…”
Section: Our Main Results Arementioning
confidence: 95%
“…It is, however, asymptotically sharp, since (∆−1) ∆−1 ∆ ∆ ∼ 1 e∆ as ∆ → ∞. For more more on zero-free regions of the hard-core partition function see [56,51].…”
Section: Convergence Of the Cluster Expansionmentioning
confidence: 99%
“…In this paper, we continue the line of research started in [Ba16a] and continued, in particular, in [Ba17], [PR17a], [Ba16b], [PR17b], [L+17], [BR17] and [EM17], on constructing efficient algorithms for computing (approximating) combinatorially defined quantities (partition functions) by exploiting the information on their complex zeros. A typical application of the method consists of a) proving that the function in question does not have zeros in some interesting domain in C n and b) constructing a low-degree polynomial approximation for the logarithm of the function in a slightly smaller domain.…”
Section: Introduction and Main Resultsmentioning
confidence: 93%
“…Usually, part a) is where the main work is done: since there is no general method to establish that a multivariate polynomial (typically having many monomials) is non-zero in a domain in C n , some quite clever arguments are being sought and found, cf. [PR17b], [EM17], see also Section 2.5 of [Ba16b] for the very few general results in this respect. Once part a) is accomplished, part b) produces a quasipolynomial approximation algorithm in a quite straightforward way, see Section 2.2 of [Ba16b].…”
Section: Introduction and Main Resultsmentioning
confidence: 99%