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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 ..."
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Cited by 61 (14 self)
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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,
Chromatic roots are dense in the whole complex plane
 In preparation
, 2000
"... to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic ..."
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Cited by 42 (15 self)
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to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Pottsmodel partition functions) ZG(q,v) outside the disc q + v  < v. An immediate corollary is that the chromatic roots of notnecessarilyplanar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof. KEY WORDS: Graph, chromatic polynomial, dichromatic polynomial, Whitney rank function, Tutte polynomial, Potts model, Fortuin–Kasteleyn representation,
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 ..."
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Cited by 17 (5 self)
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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.
Partition Function Zeros at FirstOrder Phase Transitions: A General Analysis
 COMM. MATHEMATICAL PHYSICS
, 2004
"... We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [5]. Under these assumptions, we derive equations w ..."
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Cited by 14 (2 self)
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We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [5]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions, up to an error which we prove is (generically) exponentially small in the linear size of the system. For asymptotically large systems, the zeros concentrate on phase boundaries which are simple curves ending in multiple points. For models with an Isinglike plusminus symmetry, we also establish a local version of the LeeYang Circle Theorem. This result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle, while in the complement of this region the zeros concentrate on less symmetric curves.
Absence of Zeros for the Chromatic Polynomial on Bounded Degree Graphs
, 2005
"... In this paper, I give a short proof of a recent result by Sokal, showing that all zeros of the chromatic polynomial PG(q) of a finite graph G of maximal degree D lie in the disc q  < KD, where K is a constant that is strictly smaller than 8. 1 ..."
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Cited by 12 (0 self)
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In this paper, I give a short proof of a recent result by Sokal, showing that all zeros of the chromatic polynomial PG(q) of a finite graph G of maximal degree D lie in the disc q  < KD, where K is a constant that is strictly smaller than 8. 1
Approximate counting and quantum computation
 Combinatorics, Probability and Computing
, 2006
"... Motivated by the result that an ‘approximate ’ evaluation #P have of the Jones polynomial of a braid at a 5 th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of addi ..."
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Cited by 11 (1 self)
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Motivated by the result that an ‘approximate ’ evaluation #P have of the Jones polynomial of a braid at a 5 th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P intersection and GapP have such an approximation scheme under certain natural normalizations. However we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work. 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.
Transfer matrices and partitionfunction zeros for antiferromagnetic Potts models. IV. Chromatic polynomial with . . .
, 2004
"... We study the chromatic polynomial PG(q) for m × n square and triangularlattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zerotemperature limit of the partition function for the antiferromagnetic qstate Potts model defined on the lattice G. We show how ..."
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Cited by 11 (6 self)
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We study the chromatic polynomial PG(q) for m × n square and triangularlattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zerotemperature limit of the partition function for the antiferromagnetic qstate Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin–Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex qplane in the limit n → ∞. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.
Potts Model on Infinite Graphs and the Limit of Chromatic Polynomials
 Commun. Math. Phys
, 2003
"... Given an infinite graph G quasitransitive and amenable with maximum degree #, we show that reduced ground state degeneracy per site W r (G, q) of the qstate antiferromagnetic Potts model at zero temperature on G is analytic in the variable 1/q, whenever /q < 1. This result proves, in an eve ..."
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Cited by 8 (1 self)
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Given an infinite graph G quasitransitive and amenable with maximum degree #, we show that reduced ground state degeneracy per site W r (G, q) of the qstate antiferromagnetic Potts model at zero temperature on G is analytic in the variable 1/q, whenever /q < 1. This result proves, in an even stronger formulation, a conjecture originally sketched in [12] and explicitly formulated in [16] and [19], based on which a su#cient condition for W r (G, q) to be analytic at 1/q = 0 is that G is a regular lattice. Keywords: Potts model, chromatic polynomials, cluster expansion
Zerofree Regions for Multivariate Tutte Polynomials (alias Pottsmodel Partition Functions) of Graphs and Matroids
, 2008
"... The chromatic polynomial PG(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (−∞,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all ..."
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Cited by 7 (1 self)
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The chromatic polynomial PG(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (−∞,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zerofree regions of the multivariate Tutte polynomial ZG(q,v). The proofs are quite simple, and employ deletioncontraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.