54 research outputs found
The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph
In this paper, we show that the eigenvectors of the zero Laplacian and
signless Lapacian eigenvalues of a -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{2j\pi}{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.Comment: 22 pages, 3 figure
H-Eigenvalues of Laplacian and Signless Laplacian Tensors
We propose a simple and natural definition for the Laplacian and the signless
Laplacian tensors of a uniform hypergraph. We study their H-eigenvalues,
i.e., H-eigenvalues with nonnegative H-eigenvectors, and H-eigenvalues,
i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the
Laplacian tensor, the signless Laplacian tensor and the adjacency tensor has at
most one H-eigenvalue, but has several other H-eigenvalues. We
identify their largest and smallest H-eigenvalues, and establish some
maximum and minimum properties of these H-eigenvalues. We then define
analytic connectivity of a uniform hypergraph and discuss its application in
edge connectivity
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