Spectral hypergraph theory

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Spectral hypergraph theory

 

Author: Hu, Shenglong
Title: Spectral hypergraph theory
Degree: Ph.D.
Year: 2013
Subject: Calculus of tensors.
Hong Kong Polytechnic University -- Dissertations
Department: Dept. of Applied Mathematics
Pages: xii, 107 p. : ill. ; 30 cm.
Language: English
InnoPac Record: http://library.polyu.edu.hk/record=b2652765
URI: http://theses.lib.polyu.edu.hk/handle/200/7238
Abstract: The main subject of this thesis is the study of a few basic problems in spectral hypergraph theory based on Laplacian-type tensors. These problems are hypergraph analogues of some important problems in spectral graph theory. As some foundations, we study some new problems of tensor determinant and non-negative tensor partition. Then two classes of Laplacian-type tensors for uniform hypergraphs are proposed. One is called Laplacian, and the other one Laplace-Beltrami tensor. We study the H-spectra of uniform hypergraphs through their Laplacian, and the Z-spectra of even uniform hypergraphs through their Laplace-Beltrami tensors. All the H⁺-eigenvalues of the Laplacian can be computed out through the developed partition method. Spectral component, an intrinsic notion of a uniform hypergraph, is introduced to characterize the hypergraph spectrum. Many fundamental properties of the spectrum are connected to the underlying hypergraph structures. Basic spectral hypergraph theory based on Laplacian-type tensors are built. With the theory, we study algebraic connectivity, edge connectivity, vertex connectivity, edge expansion, and spectral invariance of the hypergraph.

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