We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of developments building upon classical work has led to a rich understanding of 'symmetric hyperdeterminants' of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally. We apply this notion to the 'adjacency hypermatrix' of a uniform hypergraph, and prove a number of natural analogues of basic results in Spectral Graph Theory.
","abstract_html":"<p>We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of developments building upon classical work has led to a rich understanding of 'symmetric hyperdeterminants' of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally. We apply this notion to the 'adjacency hypermatrix' of a uniform hypergraph, and prove a number of natural analogues of basic results in Spectral Graph Theory.</p>","abstract_has_math":false,"creators":["Dutle, Aaron Michael"],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Joshua N Cooper"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T04:38:07Z","subjects":["Mathematics","Physical Sciences and Mathematics","Eigenvalue","Hypergraph","Spectrum"],"languages":[],"rights":["© 2012, Aaron Michael Dutle"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1593","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Joshua N Cooper"]},{"key":"dc:creator","label":"Author","values":["Dutle, Aaron Michael"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics","Eigenvalue","Hypergraph","Spectrum"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2012, Aaron Michael Dutle"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1593"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of developments building upon classical work has led to a rich understanding of 'symmetric hyperdeterminants' of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally. We apply this notion to the 'adjacency hypermatrix' of a uniform hypergraph, and prove a number of natural analogues of basic results in Spectral Graph Theory.
"]},{"key":"dc:title","label":"Title","values":["Spectra of Hypergraphs"]}]}],"canonical_facts":{"dc:contributor":["Joshua N Cooper"],"dc:creator":["Dutle, Aaron Michael"],"dc:description.abstract":["We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of developments building upon classical work has led to a rich understanding of 'symmetric hyperdeterminants' of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally. We apply this notion to the 'adjacency hypermatrix' of a uniform hypergraph, and prove a number of natural analogues of basic results in Spectral Graph Theory.
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