6,318 research outputs found
On the positive and negative inertia of weighted graphs
The number of the positive, negative and zero eigenvalues in the spectrum of
the (edge)-weighted graph are called positive inertia index, negative
inertia index and nullity of the weighted graph , and denoted by ,
, , respectively. In this paper, the positive and negative
inertia index of weighted trees, weighted unicyclic graphs and weighted
bicyclic graphs are discussed, the methods of calculating them are obtained.Comment: 12. arXiv admin note: text overlap with arXiv:1107.0400 by other
author
The inertia of weighted unicyclic graphs
Let be a weighted graph. The \textit{inertia} of is the triple
, where
are the number of the positive, negative and zero
eigenvalues of the adjacency matrix of including their
multiplicities, respectively. , is called the
\textit{positive, negative index of inertia} of , respectively. In this
paper we present a lower bound for the positive, negative index of weighted
unicyclic graphs of order with fixed girth and characterize all weighted
unicyclic graphs attaining this lower bound. Moreover, we characterize the
weighted unicyclic graphs of order with two positive, two negative and at
least zero eigenvalues, respectively.Comment: 23 pages, 8figure
Weighted Spectral Embedding of Graphs
We present a novel spectral embedding of graphs that incorporates weights
assigned to the nodes, quantifying their relative importance. This spectral
embedding is based on the first eigenvectors of some properly normalized
version of the Laplacian. We prove that these eigenvectors correspond to the
configurations of lowest energy of an equivalent physical system, either
mechanical or electrical, in which the weight of each node can be interpreted
as its mass or its capacitance, respectively. Experiments on a real dataset
illustrate the impact of weighting on the embedding
Euclidean Distances, soft and spectral Clustering on Weighted Graphs
We define a class of Euclidean distances on weighted graphs, enabling to
perform thermodynamic soft graph clustering. The class can be constructed form
the "raw coordinates" encountered in spectral clustering, and can be extended
by means of higher-dimensional embeddings (Schoenberg transformations).
Geographical flow data, properly conditioned, illustrate the procedure as well
as visualization aspects.Comment: accepted for presentation (and further publication) at the ECML PKDD
2010 conferenc
Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues
Let be the smallest number such that the adjacency matrix of any
undirected graph with vertices or more has at least nonpositive
eigenvalues. We show that is well-defined and prove that the values of
for are respectively. In addition, we
prove that for all , , in which
is the Ramsey number for and , and is the triangular
number. This implies new lower bounds for eigenvalues of Laplacian matrices:
the -th largest eigenvalue is bounded from below by the -th largest
degree, which generalizes some prior results.Comment: 23 pages, 12 figure
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