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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Bibliographic Details
Published inLinear algebra and its applications Vol. 436; no. 9; pp. 3268 - 3292
Main Authors Cooper, Joshua, Dutle, Aaron
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 01.05.2012
Elsevier
Subjects
Algebra
Characteristic polynomial
Exact sciences and technology
Hypergraph
Linear and multilinear algebra, matrix theory
Mathematics
Resultant
Sciences and techniques of general use
Spectrum
Hypergraph
Primary 05C65
Secondary 15A69
Resultant
Characteristic polynomial
15A18
Spectrum
Adjacency matrix
Graph theory
Integer
Algebra
Spectral theory
Eigenvalue problem
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Summary: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.
ISSN:0024-3795
DOI:10.1016/j.laa.2011.11.018