10,046 research outputs found
New graph invariants based on -Laplacian eigenvalues
We present monotonicity inequalities for certain functions involving
eigenvalues of -Laplacians on signed graphs with respect to . Inspired by
such monotonicity, we propose new spectrum-based graph invariants, called
(variational) cut-off adjacency eigenvalues, that are relevant to certain
eigenvector-dependent nonlinear eigenvalue problem. Using these invariants, we
obtain new lower bounds for the -Laplacian variational eigenvalues,
essentially giving the state-of-the-art spectral asymptotics for these
eigenvalues. Moreover, based on such invariants, we establish two inertia
bounds regarding the cardinalities of a maximum independent set and a minimum
edge cover, respectively. The first inertia bound enhances the classical
Cvetkovi\'c bound, and the second one implies that the -th -Laplacian
variational eigenvalue is of the order as tends to infinity whenever
is larger than the cardinality of a minimum edge cover of the underlying
graph. We further discover an interesting connection between graph
-Laplacian eigenvalues and tensor eigenvalues and discuss applications of
our invariants to spectral problems of tensors.Comment: 30 page
New bounds for the max--cut and chromatic number of a graph
We consider several semidefinite programming relaxations for the max--cut
problem, with increasing complexity. The optimal solution of the weakest
presented semidefinite programming relaxation has a closed form expression that
includes the largest Laplacian eigenvalue of the graph under consideration.
This is the first known eigenvalue bound for the max--cut when that is
applicable to any graph. This bound is exploited to derive a new eigenvalue
bound on the chromatic number of a graph. For regular graphs, the new bound on
the chromatic number is the same as the well-known Hoffman bound; however, the
two bounds are incomparable in general. We prove that the eigenvalue bound for
the max--cut is tight for several classes of graphs. We investigate the
presented bounds for specific classes of graphs, such as walk-regular graphs,
strongly regular graphs, and graphs from the Hamming association scheme
NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks
The graph Laplacian is a standard tool in data science, machine learning, and
image processing. The corresponding matrix inherits the complex structure of
the underlying network and is in certain applications densely populated. This
makes computations, in particular matrix-vector products, with the graph
Laplacian a hard task. A typical application is the computation of a number of
its eigenvalues and eigenvectors. Standard methods become infeasible as the
number of nodes in the graph is too large. We propose the use of the fast
summation based on the nonequispaced fast Fourier transform (NFFT) to perform
the dense matrix-vector product with the graph Laplacian fast without ever
forming the whole matrix. The enormous flexibility of the NFFT algorithm allows
us to embed the accelerated multiplication into Lanczos-based eigenvalues
routines or iterative linear system solvers and even consider other than the
standard Gaussian kernels. We illustrate the feasibility of our approach on a
number of test problems from image segmentation to semi-supervised learning
based on graph-based PDEs. In particular, we compare our approach with the
Nystr\"om method. Moreover, we present and test an enhanced, hybrid version of
the Nystr\"om method, which internally uses the NFFT.Comment: 28 pages, 9 figure
Sampling Large Data on Graphs
We consider the problem of sampling from data defined on the nodes of a
weighted graph, where the edge weights capture the data correlation structure.
As shown recently, using spectral graph theory one can define a cut-off
frequency for the bandlimited graph signals that can be reconstructed from a
given set of samples (i.e., graph nodes). In this work, we show how this
cut-off frequency can be computed exactly. Using this characterization, we
provide efficient algorithms for finding the subset of nodes of a given size
with the largest cut-off frequency and for finding the smallest subset of nodes
with a given cut-off frequency. In addition, we study the performance of random
uniform sampling when compared to the centralized optimal sampling provided by
the proposed algorithms.Comment: To be presented at GlobalSIP 201
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