Cored hypergraphs, power hypergraphs and their Laplacian H-eigenvalues

In this paper, we introduce the class of cored hypergraphs and power hypergraphs, and investigate the properties of their Laplacian H-eigenvalues. From an ordinary graph, one may generate a k-uniform hypergraph, called the kth power hypergraph of that graph. Power hypergraphs are cored hypergraphs, but not vice versa. Sunflowers, loose paths and loose cycles are power hypergraphs, while squids, generalized loose s-paths and loose s-cycles for 2≤s<k/2 are cored hypergraphs, but not power graphs in general. We show that the largest Laplacian H-eigenvalue of an even-uniform cored hypergraph is equal to its largest signless Laplacian H-eigenvalue. Especially, we find out these largest H-eigenvalues for even-uniform squids. Moreover, we show that the largest Laplacian H-eigenvalue of an odd-uniform squid, loose path and loose cycle is equal to the maximum degree, i.e., 2. We also compute the Laplacian H-spectra of the class of sunflowers. When k is odd, the Laplacian H-spectra of the loose cycle of size 3 and the loose path of length 3 are characterized as well.