9,052 research outputs found
Upper bounds on the Laplacian spread of graphs
The Laplacian spread of a graph is defined as the difference between the
largest and the second smallest eigenvalue of the Laplacian matrix of .
In this work, an upper bound for this graph invariant, that depends on first
Zagreb index, is given. Moreover, another upper bound is obtained and expressed as a function of the
nonzero coefficients of the Laplacian characteristic polynomial of a graph
Laplacian spread of graphs: lower bounds and relations with invariant parameters
The spread of an complex matrix with eigenvalues is defined by
\begin{equation*}
s\left( B\right) =\max_{i,j}\left\vert \beta _{i}-\beta _{j}\right\vert ,
\end{equation*}%
where the maximum is taken over all pairs of eigenvalues of . Let be
a graph on vertices. The concept of Laplacian spread of is defined by
the difference between the largest and the second smallest Laplacian eigenvalue
of . In this work, by combining old techniques of interlacing eigenvalues
and rank perturbation matrices new lower bounds on the Laplacian spread
of graphs are deduced, some of them involving invariant parameters of graphs,
as it is the case of the bandwidth, independence number and vertex connectivity
Strongly Regular Graphs as Laplacian Extremal Graphs
The Laplacian spread of a graph is the difference between the largest
eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the
graph. We find that the class of strongly regular graphs attains the maximum of
largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian
matrices and hence the maximum of Laplacian spreads among all simple connected
graphs of fixed order, minimum degree, maximum degree, minimum size of common
neighbors of two adjacent vertices and minimum size of common neighbors of two
nonadjacent vertices. Some other extremal graphs are also provided.Comment: 11 pages, 4 figures, 1 tabl
Bounds for different spreads of line and total graphs
In this paper we explore some results concerning the spread of the line and the total graph of a given graph.
A sufficient condition for the spread of a unicyclic graph with an odd girth to be at most the spread of its line graph is presented.
Additionally, we derive an upper bound for the spread of the line graph of graphs
on vertices having a vertex (edge) connectivity at most a positive integer .
Combining techniques of interlacing of eigenvalues, we derive lower bounds for the Laplacian and signless Laplacian spread of the total graph of a connected graph. Moreover, for a regular graph, an upper and lower bound for the spread of its total graph is given.publishe
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
- …