Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues
Abstract
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 $k$th power hypergraph of that graph. Power hypergraphs are cored hypergraphs, but not vice versa. Hyperstars, hypercycles, hyperpaths are special cases of power hypergraphs, while sunflowers are a subclass of 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 sunflowers. Moreover, we show that the largest Laplacian H-eigenvalue of an odd-uniform sunflower, hypercycle and hyperpath is equal to the maximum degree, i.e., 2. We also compute out the H-spectra of the class of hyperstars. When $k$ is odd, the H-spectra of the hypercycle of size 3 and the hyperpath of length 3 are characterized as well.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2013
- DOI:
- 10.48550/arXiv.1304.6839
- arXiv:
- arXiv:1304.6839
- Bibcode:
- 2013arXiv1304.6839H
- Keywords:
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- Mathematics - Spectral Theory;
- Mathematics - Combinatorics
- E-Print:
- 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:1304.1315