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$<sup>+$-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H$<sup>{++}$-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$<sup>{++}$-eigenvalue, but has several other H$<sup>+$-eigenvalues. We identify their largest and smallest H$<sup>+$-eigenvalues, and establish some maximum and minimum properties of these H$<sup>+$-eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.