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26
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 ..."
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Cited by 45 (11 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 pol ..."
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Cited by 35 (14 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 14 (4 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.
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 7 (0 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
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 6 (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
On rational approximation of algebraic functions
, 2006
"... Abstract. We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {rn(z)}n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual ..."
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Cited by 5 (3 self)
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Abstract. We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {rn(z)}n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall’s Conjecture for the sequence {rn(z)}n∈N in the complement CP 1 \Df, where Df is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {rn(z)}n∈N. As an application we settle the socalled 3conjecture of Egecioglu et al dealing with a 4term recursion related to a polynomial Riemann Hypothesis. 1. Introduction and
Transfer Matrices and PartitionFunction Zeros for Antiferromagnetic Potts Models
, 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 5 (4 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. Key Words: Chromatic polynomial; antiferromagnetic Potts model; triangular lattice; square lattice; transfer matrix; Fortuin–Kasteleyn representation; Beraha numbers; conformal The twodimensional Potts model conceals a very rich physics, in particular in the antiferromagnetic regime. Let G = (V, E) be a finite undirected graph with vertex set V and edge set E. The qstate Potts model is initially defined for q a positive integer in terms of
A little statistical mechanics for the graph theorist, Discrete Mathematics 310
, 2010
"... 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 5 (0 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.
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 even s ..."
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Cited by 4 (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
Certificates of factorisation for chromatic polynomials
 Electron. J. Combin
"... The chromatic polynomial gives the number of proper λcolourings of a graph G. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if P(G,λ) ..."
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Cited by 2 (2 self)
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The chromatic polynomial gives the number of proper λcolourings of a graph G. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if P(G,λ) = P(H1,λ)P(H2,λ)/P(Kr,λ) for some graphs H1 and H2 and clique Kr. It is known that the chromatic polynomial of any cliqueseparable graph, that is, a graph containing a separating rclique, has a chromatic factorisation. We show that there exist other chromatic polynomials that have chromatic factorisations but are not the chromatic polynomial of any cliqueseparable graph and identify all such chromatic polynomials of degree at most 10. We introduce the notion of a certificate of factorisation, that is, a sequence of algebraic transformations based on identities for the chromatic polynomial that explains the factorisations for a graph. We find an upper bound of n 2 2 n2 /2 for the lengths of these certificates, and find much smaller certificates for all chromatic factorisations of graphs of order ≤ 9. 1