|Title:||Spectral hypergraph theory|
|Subject:||Calculus of tensors.|
Hong Kong Polytechnic University -- Dissertations
|Department:||Department of Applied Mathematics|
|Pages:||xii, 107 p. : ill. ; 30 cm.|
|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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