19,728 research outputs found
A Cheeger Inequality for the Graph Connection Laplacian
The O(d) Synchronization problem consists of estimating a set of unknown
orthogonal transformations O_i from noisy measurements of a subset of the
pairwise ratios O_iO_j^{-1}. We formulate and prove a Cheeger-type inequality
that relates a measure of how well it is possible to solve the O(d)
synchronization problem with the spectra of an operator, the graph Connection
Laplacian. We also show how this inequality provides a worst case performance
guarantee for a spectral method to solve this problem.Comment: To appear in the SIAM Journal on Matrix Analysis and Applications
(SIMAX
The spectral analysis of random graph matrices
A random graph model is a set of graphs together with a probability distribution on that set. A random matrix is a matrix with entries consisting of random values from some specified distribution. Many different random matrices can be associated with a random graph. The spectra of these corresponding matrices are called the spectra of the random graph. The spectra of random graphs are critical to understanding the properties of random graphs. This thesis contains a number of results on the spectra and related spectral properties of several random graph models. In Chapter 1, we briefly present the background, some history as well as the main ideas behind our work. Apart from the introduction in Chapter 1, the first part of the main body of the thesis is Chapter 2. In this part we estimate the eigenvalues of the Laplacian matrix of random multipartite graphs. We also investigate some spectral properties of random multipartite graphs, such as the Laplacian energy, the Laplacian Estrada index, and the von Neumann entropy. The second part consists of Chapters 3, 4, 5 and 6. Guo and Mohar showed that mixed graphs are equivalent to digraphs if we regard (replace) each undirected edge as (by) two oppositely directed arcs. Motivated by the work of Guo and Mohar, we initially propose a new random graph model – the random mixed graph. Each arc is determined by an in-dependent random variable. The main themes of the second part are the spectra and related spectral properties of random mixed graphs. In Chapter 3, we prove that the empirical distribution of the eigenvalues of the Hermitian adjacency matrix converges to Wigner’s semicircle law. As an application of the LSD of Hermitian adjacency matrices, we estimate the Hermitian energy of a random mixed graph. In Chapter 4, we deal with the asymptotic behavior of the spectrum of the Hermitian adjacency matrix of random mixed graphs. We derive a separation result between the first and the remaining eigenvalues of the Hermitian adjacency matrix. As an application of the asymptotic behavior of the spectrum of the Hermitian adjacency matrix, we estimate the spectral moments of random mixed graphs. In Chapter 5, we prove that the empirical distribution of the eigenvalues of the normalized Hermitian Laplacian matrix converges to Wigner’s semicircle law. Moreover, in Chapter 6, we provide several results on the spectra of general random mixed graphs. In particular, we present a new probability inequality for sums of independent, random, self-adjoint matrices, and then apply this probability inequality to matrices arising from the study of random mixed graphs. We prove a concentration result involving the spectral norm of a random matrix and that of its expectation. Assuming that the probabilities of all the arcs of the mixed graph are mutually independent, we write the Hermitian adjacency matrix as a sum of random self-adjoint matrices. Using this, we estimate the spectrum of the Hermitian adjacency matrix, and prove a concentration result involving the spectrum of the normalized Hermitian Laplacian matrix and its expectation. Finally, in Chapter 7, we estimate upper bounds for the spectral radii of the skew adjacency matrix and skew Randić matrix of random oriented graphs
Convergence of spectra of graph-like thin manifolds
We consider a family of compact manifolds which shrinks with respect to an
appropriate parameter to a graph. The main result is that the spectrum of the
Laplace-Beltrami operator converges to the spectrum of the (differential)
Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On
the other hand, if the the shrinking at the vertex parts of the manifold is
sufficiently slower comparing to that of the edge parts, the limiting spectrum
corresponds to decoupled edges with Dirichlet boundary conditions at the
endpoints. At the borderline between the two regimes we have a third
possibility when the limiting spectrum can be described by a nontrivial
coupling at the vertices.Comment: 38 pages, 6 figures (small changes, Sec 9 extended
Spectra of lifted Ramanujan graphs
A random -lift of a base graph is its cover graph on the vertices
, where for each edge in there is an independent
uniform bijection , and has all edges of the form .
A main motivation for studying lifts is understanding Ramanujan graphs, and
namely whether typical covers of such a graph are also Ramanujan.
Let be a graph with largest eigenvalue and let be the
spectral radius of its universal cover. Friedman (2003) proved that every "new"
eigenvalue of a random lift of is with high
probability, and conjectured a bound of , which would be tight by
results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved
Friedman's bound to . For -regular graphs,
where and , this translates to a bound of
, compared to the conjectured .
Here we analyze the spectrum of a random -lift of a -regular graph
whose nontrivial eigenvalues are all at most in absolute value. We
show that with high probability the absolute value of every nontrivial
eigenvalue of the lift is . This result is
tight up to a logarithmic factor, and for it
substantially improves the above upper bounds of Friedman and of Linial and
Puder. In particular, it implies that a typical -lift of a Ramanujan graph
is nearly Ramanujan.Comment: 34 pages, 4 figure
Pentagrams and paradoxes
Klyachko and coworkers consider an orthogonality graph in the form of a
pentagram, and in this way derive a Kochen-Specker inequality for spin 1
systems. In some low-dimensional situations Hilbert spaces are naturally
organised, by a magical choice of basis, into SO(N) orbits. Combining these
ideas some very elegant results emerge. We give a careful discussion of the
pentagram operator, and then show how the pentagram underlies a number of other
quantum "paradoxes", such as that of Hardy.Comment: 14 pages, 4 figure
Localized behavior in the Lyapunov vectors for quasi-one-dimensional many-hard-disk systems
We introduce a definition of a "localization width" whose logarithm is given
by the entropy of the distribution of particle component amplitudes in the
Lyapunov vector. Different types of localization widths are observed, for
example, a minimum localization width where the components of only two
particles are dominant. We can distinguish a delocalization associated with a
random distribution of particle contributions, a delocalization associated with
a uniform distribution and a delocalization associated with a wave-like
structure in the Lyapunov vector. Using the localization width we show that in
quasi-one-dimensional systems of many hard disks there are two kinds of
dependence of the localization width on the Lyapunov exponent index for the
larger exponents: one is exponential, and the other is linear. Differences, due
to these kinds of localizations also appear in the shapes of the localized
peaks of the Lyapunov vectors, the Lyapunov spectra and the angle between the
spatial and momentum parts of the Lyapunov vectors. We show that the Krylov
relation for the largest Lyapunov exponent as a
function of the density is satisfied (apart from a factor) in the same
density region as the linear dependence of the localization widths is observed.
It is also shown that there are asymmetries in the spatial and momentum parts
of the Lyapunov vectors, as well as in their and -components.Comment: 41 pages, 21 figures, Manuscript including the figures of better
quality is available from http://www.phys.unsw.edu.au/~gary/Research.htm
Merging the A- and Q-spectral theories
Let be a graph with adjacency matrix , and let
be the diagonal matrix of the degrees of The signless
Laplacian of is defined as .
Cvetkovi\'{c} called the study of the adjacency matrix the %
\textit{-spectral theory}, and the study of the signless Laplacian--the
\textit{-spectral theory}. During the years many similarities and
differences between these two theories have been established. To track the
gradual change of into in this paper it
is suggested to study the convex linear combinations of and defined by This study sheds new light
on and , and yields some surprises, in
particular, a novel spectral Tur\'{a}n theorem. A number of challenging open
problems are discussed.Comment: 26 page
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