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

Chromatic roots and minor-closed families of graphs

Given a minor-closed class of graphs $\mathcal{G}$, what is the infimum of the non-trivial roots of the chromatic polynomial of $G \in \mathcal{G}$? When $\mathcal{G}$ is the class of all graphs, the answer is known to be $32/27$. We answer this question exactly for three minor-closed classes of graphs. Furthermore, we conjecture precisely when the value is larger than $32/27$.Comment: 18 pages, 5 figure

Potts Model Partition Functions for Self-Dual Families of Strip Graphs

We consider the $q$-state Potts model on families of self-dual strip graphs $G_D$ of the square lattice of width $L_y$ and arbitrarily great length $L_x$, with periodic longitudinal boundary conditions. The general partition function $Z$ and the T=0 antiferromagnetic special case $P$ (chromatic polynomial) have the respective forms $\sum_{j=1}^{N_{F,L_y,\lambda}} c_{F,L_y,j} (\lambda_{F,L_y,j})^{L_x}$, with $F=Z,P$. For arbitrary $L_y$, we determine (i) the general coefficient $c_{F,L_y,j}$ in terms of Chebyshev polynomials, (ii) the number $n_F(L_y,d)$ of terms with each type of coefficient, and (iii) the total number of terms $N_{F,L_y,\lambda}$. We point out interesting connections between the $n_Z(L_y,d)$ and Temperley-Lieb algebras, and between the $N_{F,L_y,\lambda}$ and enumerations of directed lattice animals. Exact calculations of $P$ are presented for $2 \le L_y \le 4$. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), $W(q)$. Generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine the continuous locus ${\cal B}$ in the complex $q$ plane where $W(q)$ is singular. We find the interesting result that for all $L_y$ values considered, the maximal point at which ${\cal B}$ crosses the real $q$ axis, denoted $q_c$ is the same, and is equal to the value for the infinite square lattice, $q_c=3$. This is the first family of strip graphs of which we are aware that exhibits this type of universality of $q_c$.Comment: 36 pages, latex, three postscript figure

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

Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice

We present exact solutions for the zero-temperature partition function (chromatic polynomial $P$) and the ground state degeneracy per site $W$ (= exponent of the ground-state entropy) for the $q$-state Potts antiferromagnet on strips of the square lattice of width $L_y$ vertices and arbitrarily great length $L_x$ vertices. The specific solutions are for (a) $L_y=4$, $(FBC_y,PBC_x)$ (cyclic); (b) $L_y=4$, $(FBC_y,TPBC_x)$ (M\"obius); (c) $L_y=5,6$, $(PBC_y,FBC_x)$ (cylindrical); and (d) $L_y=5$, $(FBC_y,FBC_x)$ (open), where $FBC$, $PBC$, and $TPBC$ denote free, periodic, and twisted periodic boundary conditions, respectively. In the $L_x \to \infty$ limit of each strip we discuss the analytic structure of $W$ in the complex $q$ plane. The respective $W$ functions are evaluated numerically for various values of $q$. Several inferences are presented for the chromatic polynomials and analytic structure of $W$ for lattice strips with arbitrarily great $L_y$. The absence of a nonpathological $L_x \to \infty$ limit for real nonintegral $q$ in the interval $0 < q < 3$ ($0 < q < 4$) for strips of the square (triangular) lattice is discussed.Comment: 37 pages, latex, 4 encapsulated postscript figure