68,139 research outputs found
Toward An Uncertainty Principle For Weighted Graphs
International audienceThe uncertainty principle states that a signal cannot be localized both in time and frequency. With the aim of extending this result to signals on graphs, Agaskar & Lu introduce notions of graph and spectral spreads. They show that a graph uncertainty principle holds for some families of unweighted graphs. This principle states that a signal cannot be simultaneously localized both in graph and spectral domains. In this paper, we aim to extend their work to weighted graphs. We show that a naive extension of their definitions leads to inconsistent results such as discontinuity of the graph spread when regarded as a function of the graph structure. To circumvent this problem, we propose another definition of graph spread that relies on an inverse similarity matrix. We also discuss the choice of the distance function that appears in this definition. Finally, we compute and plot uncertainty curves for families of weighted graphs
New bounds for the signless Laplacian spread
Let be an undirected simple graph. The signless Laplacian spread of is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant.publishe
New lower bounds for the RandiÄ spread
Let be an -graph. The Randi\'{c} spread of ,
, is defined as the maximum distance of its Randi\'{c} eigenvalues,
disregarding the Randi\'{c} spectral radius of . In this work, we use
numerical inequalities and bounds for the matricial spread to obtain relations
between this spectral parameter and some structural and algebraic parameters of the
underlying graph such as, the sequence of vertex degrees, the nullity, Randi\'{c} index, generalized Randi\'{c} indices and its independence number. In the last section a comparison is presented for regular graphs
On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index
The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)âRD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))â„λ2(RDL(G))â„âŠâ„λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))âλnâ1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases
A Spectral Graph Uncertainty Principle
The spectral theory of graphs provides a bridge between classical signal
processing and the nascent field of graph signal processing. In this paper, a
spectral graph analogy to Heisenberg's celebrated uncertainty principle is
developed. Just as the classical result provides a tradeoff between signal
localization in time and frequency, this result provides a fundamental tradeoff
between a signal's localization on a graph and in its spectral domain. Using
the eigenvectors of the graph Laplacian as a surrogate Fourier basis,
quantitative definitions of graph and spectral "spreads" are given, and a
complete characterization of the feasibility region of these two quantities is
developed. In particular, the lower boundary of the region, referred to as the
uncertainty curve, is shown to be achieved by eigenvectors associated with the
smallest eigenvalues of an affine family of matrices. The convexity of the
uncertainty curve allows it to be found to within by a fast
approximation algorithm requiring typically sparse
eigenvalue evaluations. Closed-form expressions for the uncertainty curves for
some special classes of graphs are derived, and an accurate analytical
approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random
graphs is developed. These theoretical results are validated by numerical
experiments, which also reveal an intriguing connection between diffusion
processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure
Signals on Graphs: Uncertainty Principle and Sampling
In many applications, the observations can be represented as a signal defined
over the vertices of a graph. The analysis of such signals requires the
extension of standard signal processing tools. In this work, first, we provide
a class of graph signals that are maximally concentrated on the graph domain
and on its dual. Then, building on this framework, we derive an uncertainty
principle for graph signals and illustrate the conditions for the recovery of
band-limited signals from a subset of samples. We show an interesting link
between uncertainty principle and sampling and propose alternative signal
recovery algorithms, including a generalization to frame-based reconstruction
methods. After showing that the performance of signal recovery algorithms is
significantly affected by the location of samples, we suggest and compare a few
alternative sampling strategies. Finally, we provide the conditions for perfect
recovery of a useful signal corrupted by sparse noise, showing that this
problem is also intrinsically related to vertex-frequency localization
properties.Comment: This article is the revised version submitted to the IEEE
Transactions on Signal Processing on May, 2016; first revision was submitted
on January, 2016; original manuscript was submitted on July, 2015. The work
includes 16 pages, 8 figure
- âŠ