Results 1  10
of
28
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
 In Survey in Combinatorics, 2005, volume 327 of London Mathematical Society Lecture Notes
, 2005
"... and matroids ..."
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros ..."
Abstract

Cited by 47 (11 self)
 Add to MetaCart
I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
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 ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
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.
Zeros of chromatic and flow polynomials of graphs
 J. Geometry
, 2003
"... We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids. 1 ..."
Abstract

Cited by 21 (4 self)
 Add to MetaCart
We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids. 1
The repulsive lattice gas, the independentset polynomial, and the Lovász local lemma
, 2004
"... We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independent ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independentset polynomial for G is nonvanishing in the polydisc of radii {px}. Furthermore, we show that the usual proof of the Lovász local lemma — which provides a sufficient condition for this to occur — corresponds to a simple inductive argument for the nonvanishing of the independentset polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37, 38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for “soft” dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternatingsign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.
Partition function zeros at firstorder phase transitions: A general analysis
 Commun. Math. Phys
, 2004
"... Abstract: This paper is a continuation of our previous analysis [2] of partition functions zeros in models with firstorder phase transitions and periodic boundary conditions. Here it is shown that the assumptions under which the results of [2] were established are satisfied by a large class of latt ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
Abstract: This paper is a continuation of our previous analysis [2] of partition functions zeros in models with firstorder phase transitions and periodic boundary conditions. Here it is shown that the assumptions under which the results of [2] were established are satisfied by a large class of lattice models. These models are characterized by two basic properties: The existence of only a finite number of ground states and the availability of an appropriate contour representation. This setting includes, for instance, the Ising, Potts and BlumeCapel models at low temperatures. The combined results of [2] and the present paper provide complete control of the zeros of the partition function with periodic boundary conditions for all models in the above class. This paper is dedicated to Elliott Lieb on the occasion of his 70 th birthday. Elliott was thesis advisor to one of us (JTC) and an inspiration to us all. AMS Subject Classification: 82B05, 82B26, 26C10, 82B20. Key words and phrases: Partition function zeros, LeeYang theorem, PirogovSinai theory, contour models. c ○ 2003 by the authors. Reproduction, by any means, of the entire article for noncommercial purposes is permitted without charge.
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 ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
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,
On the Chromatic Roots of Generalized Theta Graphs
 J. COMBINATORIAL THEORY, SERIES B
, 2000
"... The generalized theta graph \Theta s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : : ; s k 1. We prove that the roots of the chromatic polynomial (\Theta s 1 ;:::;s k ; z) of a kary generalized theta graph all lie in the disc jz \Gamma 1 ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
The generalized theta graph \Theta s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : : ; s k 1. We prove that the roots of the chromatic polynomial (\Theta s 1 ;:::;s k ; z) of a kary generalized theta graph all lie in the disc jz \Gamma 1j [1 + o(1)] k= log k, uniformly in the path lengths s i . Moreover, we prove that \Theta 2;:::;2 ' K 2;k indeed has a chromatic root of modulus [1 + o(1)] k= log k. Finally, for k 8 we prove that the generalized theta graph with a chromatic root that maximizes jz \Gamma 1j is the one with all path lengths equal to 2; we conjecture that this holds for all k.
The BrownColbourn conjecture on zeros of reliability polynomials is false
 J. Combin. Theory Ser. B
"... We give counterexamples to the Brown–Colbourn conjecture on reliability polynomials, in both its univariate and multivariate forms. The multivariate Brown–Colbourn conjecture is false already for the complete graph K4. The univariate Brown–Colbourn conjecture is false for certain simple planar graph ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
We give counterexamples to the Brown–Colbourn conjecture on reliability polynomials, in both its univariate and multivariate forms. The multivariate Brown–Colbourn conjecture is false already for the complete graph K4. The univariate Brown–Colbourn conjecture is false for certain simple planar graphs obtained from K4 by parallel and series expansion of edges. We show, in fact, that a graph has the multivariate Brown– Colbourn property if and only if it is seriesparallel.
A personal list of unsolved problems concerning Potts models and lattice gases
 TO APPEAR IN MARKOV PROCESSES AND RELATED FIELDS
, 2000
"... I review recent results and unsolved problems concerning the hardcore lattice gas and the qcoloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinitevolume Gibbs measure, complex zeros of the ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
I review recent results and unsolved problems concerning the hardcore lattice gas and the qcoloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinitevolume Gibbs measure, complex zeros of the partition function) and the dynamics of local and nonlocal Monte Carlo algorithms (ergodicity, rapid mixing, mixing at complex fugacity). These problems touch on mathematical physics, probability, combinatorics and theoretical computer science.