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.
Categories: math.SP