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dc.contributor.advisorSmyth, Chris
dc.contributor.authorTaylor, Graeme
dc.date.accessioned2011-01-26T10:27:00Z
dc.date.available2011-01-26T10:27:00Z
dc.date.issued2010
dc.identifier.urihttp://hdl.handle.net/1842/4686
dc.description.abstractWe generalise the study of cyclotomic matrices - those with all eigenvalues in the interval [-2; 2] - from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined. We introduce the notion of `4-cyclotomic' matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. Six rings OQ( p d) for d = -1;-2;-3;-7;-11;-15 give rise to examples not found in the rational-integer case; in four (d = -1;-2;-3;-7) we recover infinite families as well as sporadic cases. For d = -15;-11;-7;-2, we demonstrate that a maximal cyclotomic graph is necessarily 4- cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.en
dc.contributor.sponsorEngineering and Physical Sciences Research Council (EPSRC)en
dc.contributor.sponsorMaxwell Institute for Mathematical Sciencesen
dc.contributor.sponsorUniversity of Edinburghen
dc.contributor.sponsorEdinburgh Compute and Data Facility (ECDF)en
dc.language.isoenen
dc.publisherThe University of Edinburghen
dc.subjectcyclotomic matricesen
dc.subjectrational integeren
dc.subjectinfinite familiesen
dc.titleCyclotomic matrices and graphsen
dc.typeThesis or Dissertationen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen


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