301 research outputs found
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
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
The Euler and Springer numbers as moment sequences
I study the sequences of Euler and Springer numbers from the point of view of
the classical moment problem.Comment: LaTeX2e, 30 pages. Version 2 contains some small clarifications
suggested by a referee. Version 3 contains new footnotes 9 and 10. To appear
in Expositiones Mathematica
The falling raindrop, revisited
I reconsider the problem of a raindrop falling through mist, collecting mass,
and generalize it to allow an arbitrary power-law form for the accretion rate.
I show that the coupled differential equations can be solved by the simple
trick of temporarily eliminating time (t) in favor of the raindrop's mass (m)
as the independent variableComment: LaTex2e/revtex4, 6 page
A really simple elementary proof of the uniform boundedness theorem
I give a proof of the uniform boundedness theorem that is elementary (i.e.
does not use any version of the Baire category theorem) and also extremely
simple.Comment: LaTex2e, 5 pages. Version 2 improves the exposition by isolating the
key lemma. To appear in the American Mathematical Monthl
The Brown-Colbourn conjecture on zeros of reliability polynomials is false
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 K_4. The
univariate Brown-Colbourn conjecture is false for certain simple planar graphs
obtained from K_4 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
series-parallel.Comment: LaTeX2e, 17 pages. Version 2 makes a few small improvements in the
exposition. To appear in Journal of Combinatorial Theory
- …