{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/8224"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/8224","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Spectrum of some regular graphs with widely spaced modifications","abstract":"This thesis has two parts. The first part studies the spectrum of a family of growing trees, we show that the eigenvalues of the adjacency matrix and Laplacian matrix have high multiplicities. As the trees grow, the graphs of those eigenvalues approach a piecewise-constant \"Cantor function\", which is different from the corresponding properties of the infinite tree. The second part studies the effect of \"widely spaced\" modifications on the spectrum of some type of structured matrices. We show that by applying those modifications, new eigenvectors that are localized near the components that correspond to the modified rows appear. By knowing the approximate form of those eigenvectors, we also determine a very close (and simple) approximation to the eigenvalues, and then we show that this approximation is indeed the limit as the matrix grows.","abstract_html":"This thesis has two parts. The first part studies the spectrum of a family of growing trees, we show that the eigenvalues of the adjacency matrix and Laplacian matrix have high multiplicities. As the trees grow, the graphs of those eigenvalues approach a piecewise-constant "Cantor function", which is different from the corresponding properties of the infinite tree. The second part studies the effect of "widely spaced" modifications on the spectrum of some type of structured matrices. We show that by applying those modifications, new eigenvectors that are localized near the components that correspond to the modified rows appear. By knowing the approximate form of those eigenvectors, we also determine a very close (and simple) approximation to the eigenvalues, and then we show that this approximation is indeed the limit as the matrix grows.","abstract_has_math":false,"creators":["Liu, Xiangwei, 1976-"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Gilbert Strang."],"committee_chairs":[],"committee_members":[],"year":2001,"date_issued":"2001","date_published":"2001","updated_at":"2026-07-22T22:22:28Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/8224","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gilbert Strang."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. 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As the trees grow, the graphs of those eigenvalues approach a piecewise-constant \"Cantor function\", which is different from the corresponding properties of the infinite tree. The second part studies the effect of \"widely spaced\" modifications on the spectrum of some type of structured matrices. We show that by applying those modifications, new eigenvectors that are localized near the components that correspond to the modified rows appear. By knowing the approximate form of those eigenvectors, we also determine a very close (and simple) approximation to the eigenvalues, and then we show that this approximation is indeed the limit as the matrix grows."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Spectrum of some regular graphs with widely spaced modifications"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gilbert Strang."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["Liu, Xiangwei, 1976-"],"dc:date.accessioned":["2005-08-23T18:25:50Z"],"dc:date.available":["2005-08-23T18:25:50Z"],"dc:date.issued":["2001"],"dc:description":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.","Includes bibliographical references (p. 71-72)."],"dc:description.abstract":["This thesis has two parts. The first part studies the spectrum of a family of growing trees, we show that the eigenvalues of the adjacency matrix and Laplacian matrix have high multiplicities. As the trees grow, the graphs of those eigenvalues approach a piecewise-constant \"Cantor function\", which is different from the corresponding properties of the infinite tree. The second part studies the effect of \"widely spaced\" modifications on the spectrum of some type of structured matrices. We show that by applying those modifications, new eigenvectors that are localized near the components that correspond to the modified rows appear. By knowing the approximate form of those eigenvectors, we also determine a very close (and simple) approximation to the eigenvalues, and then we show that this approximation is indeed the limit as the matrix grows."],"dc:description.degree":["Ph.D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1721.1/8224"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Spectrum of some regular graphs with widely spaced modifications"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:28Z"}