The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph

Abstract

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a $k$-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a {\em canonical} regularization, the phases of the components of these eigenvectors only can take some uniformly distributed values in $\{\{exp}(\frac{2jπ}{k})\;|\;j\in [k]\}$. These eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors with minimal support is called {\em minimal}. The minimal canonical H-eigenvectors characterize the even (odd)-bipartite connected components of the hypergraph and vice versa, and the minimal canonical N-eigenvectors characterize some multi-partite connected components of the hypergraph and vice versa.

22 pages, 3 figures

References in corpus (2)

• The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph
• The E-Eigenvectors of Tensors

Cited by in corpus (31)

• The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph
• Spectra of general hypergraphs
• Controllability of Hypergraphs
• The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs
• H${}^{+}$-Eigenvalues of Laplacian and Signless Laplacian Tensors
• The extremal spectral radii of $k$-uniform supertrees
• M-Tensors and Nonsingular M-Tensors
• Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues
• Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue
• The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph
• MB-tensors and MB0-tensors
• Double B-tensor and quasi-double B-tensor
• Regular Uniform Hypergraphs, $s$-Cycles, $s$-Paths and Their largest Laplacian H-Eigenvalues
• Computing The Analytic Connectivity of A Uniform Hypergraph
• Spectral bipartite Turan problems on linear hypergraphs
• On the spectrum of directed uniform and non-uniform hypergraphs
• Spectral properties of general hypergraphs
• Adjacency Spectra of Random and Uniform Hypergraphs
• E-cospectral hypergraphs and some hypergraphs determined by their spectra
• On some properties of the $\alpha$-spectral radius of the $k$-uniform hypergraph
• An Even Order Symmetric B Tensor is Positive Definite
• The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices
• The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs
• Inverse Perron values and connectivity of a uniform hypergraph
• Some properties and applications of odd-colorable $r$-hypergraphs
• Multiplicities of eigenvalues of tensors
• Computing all Laplacian H-eigenvalues for a k-uniform loose path of length three
• Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs
• Minimal non-odd-transversal hypergraphs and minimal non-odd-bipartite hypergraphs
• Some spectral properties of uniform hypergraphs
• The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs