3,364 research outputs found
Invertibility of graph translation and support of Laplacian Fiedler vectors
The graph Laplacian operator is widely studied in spectral graph theory
largely due to its importance in modern data analysis. Recently, the Fourier
transform and other time-frequency operators have been defined on graphs using
Laplacian eigenvalues and eigenvectors. We extend these results and prove that
the translation operator to the 'th node is invertible if and only if all
eigenvectors are nonzero on the 'th node. Because of this dependency on the
support of eigenvectors we study the characteristic set of Laplacian
eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish
on large neighborhoods and then explicitly construct a family of non-planar
graphs that do exhibit this property.Comment: 21 pages, 7 figure
Symmetric motifs in random geometric graphs
We study symmetric motifs in random geometric graphs. Symmetric motifs are
subsets of nodes which have the same adjacencies. These subgraphs are
particularly prevalent in random geometric graphs and appear in the Laplacian
and adjacency spectrum as sharp, distinct peaks, a feature often found in
real-world networks. We look at the probabilities of their appearance and
compare these across parameter space and dimension. We then use the Chen-Stein
method to derive the minimum separation distance in random geometric graphs
which we apply to study symmetric motifs in both the intensive and
thermodynamic limits. In the thermodynamic limit the probability that the
closest nodes are symmetric approaches one, whilst in the intensive limit this
probability depends upon the dimension.Comment: 11 page
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