Spectra of uniform hypergraphs
We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of “symmetric hyperdeterminants” of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyperdeterminant...
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| Published in | Linear algebra and its applications Vol. 436; no. 9; pp. 3268 - 3292 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier Inc
01.05.2012
Elsevier |
| Subjects | |
| Online Access | Get full text |
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| Abstract | We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of recent 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 analogs of basic results in Spectral Graph Theory. Open problems abound, and we present a number of directions for further study. |
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| AbstractList | We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of recent 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 analogs of basic results in Spectral Graph Theory. Open problems abound, and we present a number of directions for further study. |
| Author | Dutle, Aaron Cooper, Joshua |
| Author_xml | – sequence: 1 givenname: Joshua surname: Cooper fullname: Cooper, Joshua email: cooper@math.sc.edu – sequence: 2 givenname: Aaron surname: Dutle fullname: Dutle, Aaron email: dutle@mailbox.sc.edu |
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| Snippet | We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work... |
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| SubjectTerms | Algebra Characteristic polynomial Exact sciences and technology Hypergraph Linear and multilinear algebra, matrix theory Mathematics Resultant Sciences and techniques of general use Spectrum |
| Title | Spectra of uniform hypergraphs |
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