1,875 research outputs found
Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits
We calculate the chromatic polynomials and, from these, the
asymptotic limiting functions
for families of -vertex graphs comprised of repeated subgraphs
adjoined to an initial graph . These calculations of for
infinitely long strips of varying widths yield important insights into
properties of for two-dimensional lattices . In turn,
these results connect with statistical mechanics, since is the
ground state degeneracy of the -state Potts model on the lattice .
For our calculations, we develop and use a generating function method, which
enables us to determine both the chromatic polynomials of finite strip graphs
and the resultant function in the limit . From
this, we obtain the exact continuous locus of points where
is nonanalytic in the complex plane. This locus is shown to
consist of arcs which do not separate the plane into disconnected regions.
Zeros of chromatic polynomials are computed for finite strips and compared with
the exact locus of singularities . We find that as the width of the
infinitely long strips is increased, the arcs comprising elongate
and move toward each other, which enables one to understand the origin of
closed regions that result for the (infinite) 2D lattice.Comment: 48 pages, Latex, 12 encapsulated postscript figures, to appear in
Physica
Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space
Associated to any finite simple graph is the chromatic polynomial
whose complex zeroes are called the chromatic zeros of .
A hierarchical lattice is a sequence of finite simple graphs
built recursively using a substitution rule
expressed in terms of a generating graph. For each , let denote the
probability measure that assigns a Dirac measure to each chromatic zero of
. Under a mild hypothesis on the generating graph, we prove that the
sequence converges to some measure as tends to infinity. We
call the limiting measure of chromatic zeros associated to
. In the case of the Diamond Hierarchical Lattice we
prove that the support of has Hausdorff dimension two.
The main techniques used come from holomorphic dynamics and more specifically
the theories of activity/bifurcation currents and arithmetic dynamics. We prove
a new equidistribution theorem that can be used to relate the chromatic zeros
of a hierarchical lattice to the activity current of a particular marked point.
We expect that this equidistribution theorem will have several other
applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added
considerably more background on activity currents and especially on the
Dujardin-Favre classification of the passive locus. Exposition in the proof
of the main theorem was improved. Comments welcome
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P_G(q) for the generalized
theta graphs \Theta^{(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 Potts-model
partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate
corollary is that the chromatic zeros of not-necessarily-planar 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.Comment: LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3
adds a new Theorem 1.4 and a new Section 5, and makes several small
improvements. To appear in Combinatorics, Probability & Computin
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
Ground State Entropy of the Potts Antiferromagnet on Cyclic Strip Graphs
We present exact calculations of the zero-temperature partition function
(chromatic polynomial) and the (exponent of the) ground-state entropy for
the -state Potts antiferromagnet on families of cyclic and twisted cyclic
(M\"obius) strip graphs composed of -sided polygons. Our results suggest a
general rule concerning the maximal region in the complex plane to which
one can analytically continue from the physical interval where . The
chromatic zeros and their accumulation set exhibit the rather
unusual property of including support for and provide further
evidence for a relevant conjecture.Comment: 7 pages, Latex, 4 figs., J. Phys. A Lett., in pres
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