Spectra of Hypergraphs

<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>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Campus Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2012

Author dc:creator

• Dutle, Aaron Michael

Contributors dc:contributor

• Joshua N Cooper

• Mathematics
• Physical Sciences and Mathematics
• Eigenvalue
• Hypergraph
• Spectrum

Statement dc:rights

• © 2012, Aaron Michael Dutle

Repository record dc:identifier

https://scholarcommons.sc.edu/etd/1593

OAI identifier oai:identifier

oai:scholarcommons.sc.edu:etd-2594