5 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
Expedition in Data and Harmonic Analysis on Graphs
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 i’th node is invertible if and only if all eigenvectors are nonzero on the i’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.
We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time