Results 1  10
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77
Transfer matrices and partitionfunction zeros for antiferromagnetic Potts models. I. General theory and squarelattice chromatic polynomial
 J. Stat. Phys
, 2001
"... We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer ..."
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Cited by 33 (6 self)
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We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the FortuinKasteleyn representation. In particular, we extract the limiting curves of partitionfunction zeros when n → ∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B2,B3,B4,B5 are limiting points of partitionfunction zeros as n → ∞ whenever the strip width m is ≥ 7 (periodic transverse b.c.) or ≥ 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B10) cannot be a chromatic root of any graph. Key Words: Chromatic polynomial; chromatic root; antiferromagnetic Potts model; square lattice; transfer matrix; FortuinKasteleyn representation; TemperleyLieb algebra; Beraha–Kahane–Weiss theorem; Beraha numbers.
Negative dependence and the geometry of polynomials
, 2008
"... We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures, uniform random spanning tree measur ..."
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Cited by 22 (10 self)
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We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures, uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons’ recent results on determinantal measures and we construct counterexamples to several conjectures of Pemantle
Knot Invariants and the BollobasRiordan Polynomial of embedded graphs, preprint arXiv:math.CO/0605466. N. Reshetikhin and V. Turaev, Ribbon graphs and their invariants derived from quantum groups
 Comm. Math. Phys
, 1990
"... Abstract. For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ × I. We relate the HOMFLY polynomial of L(G) to the recently defined BollobásRiordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We u ..."
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Cited by 16 (6 self)
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Abstract. For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ × I. We relate the HOMFLY polynomial of L(G) to the recently defined BollobásRiordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot theory to prove results about graph polynomials and, after discussing questions of equivalence of the polynomials, we go on to use our formulae to prove a duality relation for the BollobásRiordan polynomial. We then consider the specialization to the Jones polynomial and recent results of Chmutov and Pak to relate the BollobásRiordan polynomials of an embedded graph and its tensor product with a cycle. 1.
The LeeYang and PólyaSchur programs I. Linear operators preserving stability
, 2008
"... In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving nonvanishing properties are an essential tool in this program as well as various contexts in complex analysis, probability theory, combinatorics, matrix t ..."
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Cited by 14 (8 self)
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In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving nonvanishing properties are an essential tool in this program as well as various contexts in complex analysis, probability theory, combinatorics, matrix theory. We characterize all linear operators on finite or infinitedimensional spaces of multivariate polynomials preserving the property of being nonvanishing when the variables are in prescribed open circular domains. In particular, this supersedes [7, 9] and solves the higher dimensional counterpart of a longstanding classification problem originating from classical works of Hermite, Laguerre,
Generalized duality for graphs on surfaces and the signed BollobasRiordan polynomial
 J. Combin. Theory Ser. B
"... Abstract. We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed BollobásRiordan polynomials of dual graphs. This relation unifies various r ..."
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Cited by 14 (0 self)
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Abstract. We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed BollobásRiordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the BollobásRiordan polynomials.
Negatively correlated random variables and Mason’s conjecture
 Department of Mathematics, Stockholm University
, 2006
"... Abstract. Mason’s Conjecture asserts that for an m–element rank r matroid M the sequence (Ik / � � m k: 0 ≤ k ≤ r) is logarithmically concave, in which Ik is the number of independent k–sets of M. A related conjecture in probability theory implies these inequalities provided that the set of indepe ..."
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Cited by 13 (3 self)
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Abstract. Mason’s Conjecture asserts that for an m–element rank r matroid M the sequence (Ik / � � m k: 0 ≤ k ≤ r) is logarithmically concave, in which Ik is the number of independent k–sets of M. A related conjecture in probability theory implies these inequalities provided that the set of independent sets of M satisfies a strong negative correlation property we call the Rayleigh condition. This condition is known to hold for the set of bases of a regular matroid. We show that if ω is a weight function on a set system Q that satisfies the Rayleigh condition then Q is a convex delta–matroid and ω is logarithmically submodular. Thus, the hypothesis of the probabilistic conjecture leads inevitably to matroid theory. We also show that two–sums of matroids preserve the Rayleigh condition in four distinct senses, and hence that the Potts model of an iterated two–sum of uniform matroids satisfies
From a zoo to a zoology: Towards a general theory of graph polynomials
 Theory of Computing Systems
, 2007
"... Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We in ..."
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Cited by 13 (4 self)
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Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities. 1
COMBINATORICS AND GEOMETRY OF POWER IDEALS
, 2009
"... We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We pr ..."
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Cited by 12 (1 self)
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We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A, and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results due to DahmenMicchelli, HoltzRon, PostnikovShapiroShapiro, and SturmfelsXu, among others. It also settles a conjecture of HoltzRon on the spline interpolation of functions on the lattice points of a zonotope.
NCOTS (National Census Office for the Tertiary Sector
 The First Census on the Tertiary Industry in China: Summary Statistics, China Statistical
, 1996
"... Abstract. 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 poly ..."
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Cited by 11 (2 self)
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Abstract. 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. 1
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 11 (2 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.