Spectra of Uniform Hypergraphs
By Joshua N. Cooper, Aaron Dutle. 2011
524 citations recorded by the source corpus.
We present a spectral theory of hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of "hyperdeterminants" of hypermatrices, a.k.a. multidimensional arrays. 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. Open problems abound, and we present a number of directions for further study.
Subfields: Computational Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/91572dea9ddeac1344ec529a4a9dbe79404ea5d8
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Multilayer networks (2013, background): “This connection is useful, for example, for investigating spectral theory in adjacency tensors of hypergraphs [87,263,279].”
- EIGENVALUES AND LINEAR QUASIRANDOM HYPERGRAPHS (2012, methodology): “There have been three independently developed approaches to hypergraph eigenvalues: a definition by Chung [11] and Lu and Peng [34, 35] using matrices, an approach of Friedman and Wigderson [21, 22] and Cooper and Dutle [17], and lastly the eigenvalues of the shadow graph [7, 19, 36, 37, 39].”
- Networks beyond pairwise interactions: structure and dynamics (2020, methodology): “Then, one introduces a Laplacian tensor which is the discretization of the higher order Laplace-Beltrami operator in differential geometry [138, 139].”
- The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph (2013, background): “The trivial hyperstar is the one edge hypergraph, its spectrum is very clear [5].”
- Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues (2013, background): “For discussion on eigenvalues of adjacency tensors and the other types of Laplacian tensors of k-uniform hypergraphs, see [4,7,11,12,16,17,19,20] and references therein.”
- H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors (2013, background): “1 of [6] concerns the spectrum of the union of two disjoint hypergraphs.”
- Symmetric nonnegative tensors and copositive tensors (2012, methodology): “Cooper and Dutle [4] established (2) for the adjacency tensor of a connected uniform hypergraph.”
- M-Tensors and Some Applications (2014, unknown): “Tensors are increasingly ubiquitous in various areas of applied, computational, and industrial mathematics and have wide applications in data analysis and mining, information science, signal/image processing, and computational biology as well [5, 9, 15, 17].”
- Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs (2014, background): “1 ([8], The Cartesian product of hypergraphs): Let G and H be two hypergraphs.”
- Some upper bounds on the eigenvalues of uniform hypergraphs (2015, unknown): “(See [3].”
- Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process (2013, background): “In the recent years, a number of papers appeared in spectral hypergraph theory via tensors [4, 11, 12, 13, 14, 16, 19, 22, 23, 31, 30].”
- On the spectrum of hypergraphs (2017, background): “Using characteristic polynomial, the spectrum of the adjacency matrix of a graph is extended for uniform hypergraphs in [15].”