17,878 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
A Weighted Correlation Index for Rankings with Ties
Understanding the correlation between two different scores for the same set
of items is a common problem in information retrieval, and the most commonly
used statistics that quantifies this correlation is Kendall's . However,
the standard definition fails to capture that discordances between items with
high rank are more important than those between items with low rank. Recently,
a new measure of correlation based on average precision has been proposed to
solve this problem, but like many alternative proposals in the literature it
assumes that there are no ties in the scores. This is a major deficiency in a
number of contexts, and in particular while comparing centrality scores on
large graphs, as the obvious baseline, indegree, has a very large number of
ties in web and social graphs. We propose to extend Kendall's definition in a
natural way to take into account weights in the presence of ties. We prove a
number of interesting mathematical properties of our generalization and
describe an algorithm for its computation. We also validate the
usefulness of our weighted measure of correlation using experimental data
Spectral reciprocity and matrix representations of unbounded operators
Motivated by potential theory on discrete spaces, we study a family of
unbounded Hermitian operators in Hilbert space which generalize the usual
graph-theoretic discrete Laplacian. These operators are discrete analogues of
the classical conformal Laplacians and Hamiltonians from statistical mechanics.
For an infinite discrete set , we consider operators acting on Hilbert
spaces of functions on , and their representations as infinite matrices; the
focus is on , and the energy space . In
particular, we prove that these operators are always essentially self-adjoint
on , but may fail to be essentially self-adjoint on
. In the general case, we examine the von Neumann
deficiency indices of these operators and explore their relevance in
mathematical physics. Finally we study the spectra of the
operators with the use of a new approximation
scheme.Comment: 20 pages, 1 figure. To appear: Journal of Functional Analysi
Graphs of bounded degree and the -harmonic boundary
Let be a real number greater than one and let be a connected graph of
bounded degree. In this paper we introduce the -harmonic boundary of . We
use this boundary to characterize the graphs for which the constant
functions are the only -harmonic functions on . It is shown that any
continuous function on the -harmonic boundary of can be extended to a
function that is -harmonic on . Some properties of this boundary that are
preserved under rough-isometries are also given. Now let be a finitely
generated group. As an application of our results we characterize the vanishing
of the first reduced -cohomology of in terms of the
cardinality of its -harmonic boundary. We also study the relationship
between translation invariant linear functionals on a certain difference space
of functions on , the -harmonic boundary of with the first
reduced -cohomology of .Comment: Give a new proof for theorem 4.7. Change the style of the text in the
first two section
Dirichlet to Neumann Maps for Infinite Quantum Graphs
The Dirichlet problem and Dirichlet to Neumann map are analyzed for elliptic
equations on a large collection of infinite quantum graphs. For a dense set of
continuous functions on the graph boundary, the Dirichlet to Neumann map has
values in the Radon measures on the graph boundary
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