Journal
LINEAR ALGEBRA AND ITS APPLICATIONS
Volume 436, Issue 9, Pages 3268-3292Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.laa.2011.11.018
Keywords
Hypergraph; Spectrum; Resultant; Characteristic polynomial
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Funding
- NSF [DMS-1001370]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1001370] Funding Source: National Science Foundation
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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. (C) 2011 Elsevier Inc. All rights reserved.
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