1,745 research outputs found
Chromatic polynomials
In this thesis, we shall investigate chromatic polynomials of graphs, and some related polynomials. In Chapter 1, we study the chromatic polynomial written in a modified form, and use these results to characterise the chromatic polynomials of polygon trees. In Chapter 2, we consider the chromatic polynomial written as a sum of the chromatic polynomials of complete graphs; in particular, we determine for which graphs the coefficients are symmetrical, and show that the coefficients exhibit a skewed property. In Chapter 3, we dualise many results about chromatic polynomials to flow polynomials, including the results in Chapter 1, and a result about a zero-free interval. Finally, in Chapter 4, we investigate the zeros of the Tutte Polynomial; in particular their observed proximity to certain hyperbole in the xy-plane
Chromatic polynomials
In this thesis, we shall investigate chromatic polynomials of graphs, and some related polynomials. In Chapter 1, we study the chromatic polynomial written in a modified form, and use these results to characterise the chromatic polynomials of polygon trees. In Chapter 2, we consider the chromatic polynomial written as a sum of the chromatic polynomials of complete graphs; in particular, we determine for which graphs the coefficients are symmetrical, and show that the coefficients exhibit a skewed property. In Chapter 3, we dualise many results about chromatic polynomials to flow polynomials, including the results in Chapter 1, and a result about a zero-free interval. Finally, in Chapter 4, we investigate the zeros of the Tutte Polynomial; in particular their observed proximity to certain hyperbole in the xy-plane
A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees
It is proved that if is a graph containing a spanning tree with at most
three leaves, then the chromatic polynomial of has no roots in the interval
, where is the smallest real root of the
polynomial . We also construct a family of
graphs containing such spanning trees with chromatic roots converging to
from above. We employ the Whitney -switch operation to manage the analysis
of an infinite class of chromatic polynomials.Comment: 16 pages, 5 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
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
Potts Model Partition Functions for Self-Dual Families of Strip Graphs
We consider the -state Potts model on families of self-dual strip graphs
of the square lattice of width and arbitrarily great length ,
with periodic longitudinal boundary conditions. The general partition function
and the T=0 antiferromagnetic special case (chromatic polynomial) have
the respective forms , with . For arbitrary , we determine (i)
the general coefficient in terms of Chebyshev polynomials, (ii)
the number of terms with each type of coefficient, and (iii) the
total number of terms . We point out interesting connections
between the and Temperley-Lieb algebras, and between the
and enumerations of directed lattice animals. Exact
calculations of are presented for . In the limit of
infinite length, we calculate the ground state degeneracy per site (exponent of
the ground state entropy), . Generalizing from to
, we determine the continuous locus in the complex
plane where is singular. We find the interesting result that for all
values considered, the maximal point at which crosses the real
axis, denoted is the same, and is equal to the value for the infinite
square lattice, . This is the first family of strip graphs of which we
are aware that exhibits this type of universality of .Comment: 36 pages, latex, three postscript figure
On Zero-free Intervals of Flow Polynomials
This article studies real roots of the flow polynomial of a
bridgeless graph . For any integer , let be the supremum in
such that has no real roots in for all
graphs with , where is the set of vertices in of
degrees larger than . We prove that can be determined by considering
a finite set of graphs and show that for ,
, and . We also prove
that for any bridgeless graph , if all roots of are
real but some of these roots are not in the set , then and has at least 9 real roots in .Comment: 26 pages, 7 figure
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