55 research outputs found
Rayleigh-Ritz majorization error bounds of the mixed type
The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with
respect to vectors is bounded in terms of the norms of the residual vectors and
the angle between vectors in [\doi{10.1137/120884468}]. We substitute
multidimensional subspaces for the vectors and derive new bounds of absolute
changes of eigenvalues of the matrix RQ in terms of singular values of residual
matrices and principal angles between subspaces, using majorization. We show
how our results relate to bounds for eigenvalues after discarding off-diagonal
blocks or additive perturbations.Comment: 20 pages, 1 figure. Accepted to SIAM Journal on Matrix Analysis and
Application
Normalized graph Laplacians for directed graphs
We consider the normalized Laplace operator for directed graphs with positive
and negative edge weights. This generalization of the normalized Laplace
operator for undirected graphs is used to characterize directed acyclic graphs.
Moreover, we identify certain structural properties of the underlying graph
with extremal eigenvalues of the normalized Laplace operator. We prove
comparison theorems that establish a relationship between the eigenvalues of
directed graphs and certain undirected graphs. This relationship is used to
derive eigenvalue estimates for directed graphs. Finally we introduce the
concept of neighborhood graphs for directed graphs and use it to obtain further
eigenvalue estimates.Comment: 40 pages, 3 figure
An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification
We prove the following Alon-Boppana type theorem for general (not necessarily
regular) weighted graphs: if is an -node weighted undirected graph of
average combinatorial degree (that is, has edges) and girth , and if are the
eigenvalues of the (non-normalized) Laplacian of , then (The Alon-Boppana theorem implies that if is unweighted and
-regular, then if the diameter is at least .)
Our result implies a lower bound for spectral sparsifiers. A graph is a
spectral -sparsifier of a graph if where is the Laplacian matrix of and is
the Laplacian matrix of . Batson, Spielman and Srivastava proved that for
every there is an -sparsifier of average degree where
and the edges of are a
(weighted) subset of the edges of . Batson, Spielman and Srivastava also
show that the bound on cannot be reduced below when is a clique; our Alon-Boppana-type result implies that
cannot be reduced below when comes
from a family of expanders of super-constant degree and super-constant girth.
The method of Batson, Spielman and Srivastava proves a more general result,
about sparsifying sums of rank-one matrices, and their method applies to an
"online" setting. We show that for the online matrix setting the bound is tight, up to lower order terms
Graphs with Given Degree Sequence and Maximal Spectral Radius
We describe the structure of those graphs that have largest spectral radius
in the class of all connected graphs with a given degree sequence. We show that
in such a graph the degree sequence is non-increasing with respect to an
ordering of the vertices induced by breadth-first search. For trees the
resulting structure is uniquely determined up to isomorphism. We also show that
the largest spectral radius in such classes of trees is strictly monotone with
respect to majorization.Comment: 12 pages, 4 figures; revised version. Important change: Theorem 3
(formely Theorem 7) now states (and correctly proofs) the majorization result
only for "degree sequences of trees" (instead for general connected graphs).
Bo Zhou from the South China Normal University in Guangzhou, P.R. China, has
found a counter-example to the stronger resul
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