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.