Expedition in Data and Harmonic Analysis on Graphs

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dc.contributor.advisorOkoudjou, Kasso Aen_US
dc.contributor.advisorBenedetto, John Jen_US
dc.contributor.authorBegué, Matthew Josephen_US
dc.contributor.departmentMathematicsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2016-06-22T05:59:13Z
dc.date.available2016-06-22T05:59:13Z
dc.date.issued2016en_US
dc.description.abstractThe 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.en_US
dc.identifierhttps://doi.org/10.13016/M22R30
dc.identifier.urihttp://hdl.handle.net/1903/18291
dc.language.isoenen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledEigenvectorsen_US
dc.subject.pquncontrolledFourier Analysisen_US
dc.subject.pquncontrolledGraphen_US
dc.subject.pquncontrolledHarmonic Analysisen_US
dc.subject.pquncontrolledLaplacianen_US
dc.subject.pquncontrolledSpectral Graph Theoryen_US
dc.titleExpedition in Data and Harmonic Analysis on Graphsen_US
dc.typeDissertationen_US

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