H${}^{+}$-Eigenvalues of Laplacian and Signless Laplacian Tensors

Abstract

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.

References in corpus (3)

• The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph
• The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph
• On spectral hypergraph theory of the adjacency tensor

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• Doubly Nonnegative Tensors, Completely Positive Tensors and Applications
• Analytic connectivity of $k$-uniform hypergraphs
• Symmetric Nonnegative Tensors and Copositive Tensors