39 research outputs found
The Brown-Colbourn conjecture on zeros of reliability polynomials is false
We give counterexamples to the Brown-Colbourn conjecture on reliability
polynomials, in both its univariate and multivariate forms. The multivariate
Brown-Colbourn conjecture is false already for the complete graph K_4. The
univariate Brown-Colbourn conjecture is false for certain simple planar graphs
obtained from K_4 by parallel and series expansion of edges. We show, in fact,
that a graph has the multivariate Brown-Colbourn property if and only if it is
series-parallel.Comment: LaTeX2e, 17 pages. Version 2 makes a few small improvements in the
exposition. To appear in Journal of Combinatorial Theory
Exact two-terminal reliability of some directed networks
The calculation of network reliability in a probabilistic context has long
been an issue of practical and academic importance. Conventional approaches
(determination of bounds, sums of disjoint products algorithms, Monte Carlo
evaluations, studies of the reliability polynomials, etc.) only provide
approximations when the network's size increases, even when nodes do not fail
and all edges have the same reliability p. We consider here a directed, generic
graph of arbitrary size mimicking real-life long-haul communication networks,
and give the exact, analytical solution for the two-terminal reliability. This
solution involves a product of transfer matrices, in which individual
reliabilities of edges and nodes are taken into account. The special case of
identical edge and node reliabilities (p and rho, respectively) is addressed.
We consider a case study based on a commonly-used configuration, and assess the
influence of the edges being directed (or not) on various measures of network
performance. While the two-terminal reliability, the failure frequency and the
failure rate of the connection are quite similar, the locations of complex
zeros of the two-terminal reliability polynomials exhibit strong differences,
and various structure transitions at specific values of rho. The present work
could be extended to provide a catalog of exactly solvable networks in terms of
reliability, which could be useful as building blocks for new and improved
bounds, as well as benchmarks, in the general case
Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan
The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the
generalized fan have been calculated exactly for arbitrary size as well as
arbitrary individual edge and node reliabilities, using transfer matrices of
dimension four at most. While the all-terminal reliabilities of these graphs
are identical, the special case of identical edge () and node ()
reliabilities shows that their two-terminal reliabilities are quite distinct,
as demonstrated by their generating functions and the locations of the zeros of
the reliability polynomials, which undergo structural transitions at
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of 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 of
second-largest degree , the zeros of lie in the disc . 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