466,083 research outputs found
Probability of graphs with large spectral gap by multicanonical Monte Carlo
Graphs with large spectral gap are important in various fields such as
biology, sociology and computer science. In designing such graphs, an important
question is how the probability of graphs with large spectral gap behaves. A
method based on multicanonical Monte Carlo is introduced to quantify the
behavior of this probability, which enables us to calculate extreme tails of
the distribution. The proposed method is successfully applied to random
3-regular graphs and large deviation probability is estimated.Comment: 3pages 4figure
Estimation of Laplacian spectra of direct and strong product graphs
Calculating a product of multiple graphs has been studied in mathematics,
engineering, computer science, and more recently in network science,
particularly in the context of multilayer networks. One of the important
questions to be addressed in this area is how to characterize spectral
properties of a product graph using those of its factor graphs. While several
such characterizations have already been obtained analytically (mostly for
adjacency spectra), characterization of Laplacian spectra of direct product and
strong product graphs has remained an open problem. Here we develop practical
methods to estimate Laplacian spectra of direct and strong product graphs from
spectral properties of their factor graphs using a few heuristic assumptions.
Numerical experiments showed that the proposed methods produced reasonable
estimation with percentage errors confined within a +/-10% range for most
eigenvalues.Comment: 14 pages, 7 figures; to be published in Discrete Applied Mathematic
The normalized Laplacian spectrum of subdivisions of a graph
Determining and analyzing the spectra of graphs is an important and exciting research topic in mathematics science and theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.Postprint (published version
New results on word-representable graphs
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
for each . The set of word-representable graphs
generalizes several important and well-studied graph families, such as circle
graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at
most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and
A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202.
Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in
Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.],
in the present paper we show that not all graphs of vertex degree at most 4 are
word-representable. Combining this result with some previously known facts, we
derive that the number of -vertex word-representable graphs is
Concentration of measure and mixing for Markov chains
We consider Markovian models on graphs with local dynamics. We show that,
under suitable conditions, such Markov chains exhibit both rapid convergence to
equilibrium and strong concentration of measure in the stationary distribution.
We illustrate our results with applications to some known chains from computer
science and statistical mechanics.Comment: 28 page
Eigenvectors of random matrices: A survey
Eigenvectors of large matrices (and graphs) play an essential role in
combinatorics and theoretical computer science. The goal of this survey is to
provide an up-to-date account on properties of eigenvectors when the matrix (or
graph) is random.Comment: 64 pages, 1 figure; added Section 7 on localized eigenvector
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