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139
The multivariate signed BollobásRiordan polynomial
, 2008
"... We generalise the signed BollobásRiordan polynomial of S. Chmutov and I. Pak [1] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [2] and show that the duality transformation of the multivariate Tut ..."
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Cited by 19 (4 self)
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We generalise the signed BollobásRiordan polynomial of S. Chmutov and I. Pak [1] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [2] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.
The LeeYang and PólyaSchur programs. II. Theory of stable polynomials and applications
, 2008
"... In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pó ..."
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Cited by 19 (5 self)
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In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by PólyaSchur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with LeeYang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. in
ON THE COMPLEXITY OF THE INTERLACE POLYNOMIAL
, 2008
"... We consider the twovariable interlace polynomial introduced by Arratia, Bollobás and Sorkin (2004). We develop two graph transformations which allow us to derive pointtopoint reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational co ..."
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Cited by 17 (3 self)
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We consider the twovariable interlace polynomial introduced by Arratia, Bollobás and Sorkin (2004). We develop two graph transformations which allow us to derive pointtopoint reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #Phard to evaluate at every point of the plane, except at one line, where it is trivially polynomial time computable, and four lines and two points, where the complexity mostly is still open. This solves a problem posed by Arratia, Bollobás and Sorkin (2004). In particular, we observe that three specializations of the twovariable interlace polynomial, the vertexnullity interlace polynomial, the vertexrank interlace polynomial and the independent set polynomial, are almost everywhere #Phard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at every point except at −1 and 0.
A multivariate interlace polynomial and its computation for graphs
"... of bounded cliquewidth ..."
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Computing Tutte Polynomials
 THE FIRST CENSUS ON THE TERTIARY INDUSTRY IN CHINA: SUMMARY STATISTICS, CHINA STATISTICAL
, 1996
"... The Tutte polynomial of a graph, also known as the partition function of the qstate Potts model, is a 2variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial an ..."
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Cited by 16 (1 self)
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The Tutte polynomial of a graph, also known as the partition function of the qstate Potts model, is a 2variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants such as the number of spanning trees as complete evaluations. However despite its ubiquity, there are no widelyavailable effective computational tools able to compute the Tutte polynomial of a general graph of reasonable size. In this paper we describe the implementation of a program that exploits isomorphisms in the computation tree to extend the range of graphs for which it is feasible to compute their Tutte polynomials. We also consider edgeselection heuristics which give good performance in practice. We empirically demonstrate the utility of our program on random graphs. More evidence of its usefulness arises from our success in finding counterexamples to a conjecture of Welsh on the location of the real flow roots of a graph.
A criterion for the halfplane property
 DISCRETE MATH
, 2007
"... We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the halfplane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [5]. ..."
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Cited by 13 (1 self)
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We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the halfplane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [5].
Polynomial invariants of graphs and totally categorical theories
 MODNET Preprint No
, 2006
"... Abstract. In the analysis of the structure of totally categorical first order theories, the second author showed that certain combinatorial counting functions play an important role. Those functions are invariants of the structures and are always polynomials in one or many variables, depending on th ..."
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Cited by 11 (7 self)
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Abstract. In the analysis of the structure of totally categorical first order theories, the second author showed that certain combinatorial counting functions play an important role. Those functions are invariants of the structures and are always polynomials in one or many variables, depending on the number of independent dimensions of the theory in question. The first author introduced the notion of graph polynomials definable in Monadic Second Order Logic, and showed that the Tutte polynomial and its generalization, the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. This definition can be extended to allow definability in full second order, or even higher order Logic. The purpose of this paper is to show that many graph polynomials and combinatorial counting functions of graph theory do occur as combinatorial counting functions of totally categorical theory. We also give a characterization of polynomials definable in Second Order Logic. 1.
A little statistical mechanics for the graph theorist
, 2008
"... In this survey, we give a friendly introduction from a graph theory perspective to the qstate Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalen ..."
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Cited by 11 (2 self)
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In this survey, we give a friendly introduction from a graph theory perspective to the qstate Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zerotemperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches.