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Spectra of Uniform Hypergraphs
Joshua Cooper,Aaron Dutle +1 more
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TLDR
In this article, a spectral theory of hypergraphs is presented, which closely parallels Spectral Graph Theory, and it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally.Abstract:
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.read more
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On spectral hypergraph theory of the adjacency tensor
Kelly J. Pearson,Tan Zhang +1 more
TL;DR: In this paper, the adjacency tensor of a uniform multi-hypergraph is studied and conditions for the largest positive eigenvalue corresponding to a strictly positive vector are given.
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Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms
TL;DR: It is shown that there can be no linear operator for hypergraphs whose spectra captures hypergraph expansion in a Cheeger-like manner, and the Laplacian operator introduced is non-linear, and thus computing its eigenvalues exactly is intractable.
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The Spectral Theory of Tensors (Rough Version)
TL;DR: The spectral theory of tensors is an important part of numerical multi-linear algebra, or tensor computation as mentioned in this paper, and has attracted attention of mathematicians from different disciplines, such as automatic control, statistical data analysis, optimization, magnetic resonance imaging, solid mechanics, quantum physics, higher order Markov chains, spectral hypergraph theory, Finsler geometry, etc.
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Eigenvalues and Linear Quasirandom Hypergraphs
John Lenz,Dhruv Mubayi +1 more
TL;DR: In this paper, the authors connect previous notions on linear hypergraph quasirandomness of Kohayakawa-R\"odl-Skokan and Conlon-H\`{a}n-Person-Schacht and the spectral approach of Friedman-Wigderson.
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Spectral Radius on Linear $r$-Graphs without Expanded $K_{r+1}$
Guorong Gao,An Chang,Yuan Hou +2 more
TL;DR: In this article , it was shown that the spectral radius of the adjacency tensor of a transversal hypergraph is no more than Θ(n 2 ) for sufficiently large polygonal numbers.
References
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Singular values and eigenvalues of tensors: a variational approach
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Algebraic connectivity of an even uniform hypergraph
Shenglong Hu,Liqun Qi +1 more
TL;DR: Algebraic connectivity of an even uniform hypergraph based on Z-eigenvalues of the corresponding Laplacian tensor is introduced and its connections with edge connectivity and vertex connectivity are discussed.
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TL;DR: In this paper, the authors present the Unicursal Paths in a Network of Degree 4, a network of degree 4, where each degree corresponds to a node in a graph.