9 research outputs found
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
Algebraic methods for chromatic polynomials
The chromatic polynomials of certain families of graphs can be calculated by a transfer matrix method. The transfer matrix commutes with an action of the symmetric group on the colours. Using representation theory, it is shown that the matrix is equivalent to a block-diagonal matrix. The multiplicities and the sizes of the blocks are obtained.
Using a repeated inclusion-exclusion argument the entries of the blocks can be
calculated. In particular, from one of the inclusion-exclusion arguments it follows
that the transfer matrix can be written as a linear combination of operators which,
in certain cases, form an algebra. The eigenvalues of the blocks can be inferred
from this structure.
The form of the chromatic polynomials permits the use of a theorem by Beraha,
Kahane and Weiss to determine the limiting behaviour of the roots. The theorem
says that, apart from some isolated points, the roots approach certain curves in the
complex plane. Some improvements have been made in the methods of calculating
these curves.
Many examples are discussed in detail. In particular the chromatic polynomials
of the family of the so-called generalized dodecahedra and four similar families of
cubic graphs are obtained, and the limiting behaviour of their roots is discussed
A matrix method for chromatic polynomials
AbstractThe chromatic polynomials of certain families of graphs can be expressed in terms of the eigenspaces of a linear operator. The operator is represented by a matrix, which is referred to here as the compatibility matrix. In this paper complete sets of eigenfunctions are obtained for several related families, and the results are used to provide information about the location of the zeros of the associated chromatic polynomials. A number of uniform features are observed, and these are explained in terms of general properties of the underlying construction