Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ 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 $G$ of second-largest degree $\le r$, the zeros of $P_G(q)$ lie in the disc $|q| < C(r) + 1$. 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

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 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