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`http://hdl.handle.net/10397/16260`

Title: | Regular uniform hypergraphs, s-cycles, s-paths and their largest Laplacian H-eigenvalues |

Authors: | Qi, L Shao, JY Wang, Q |

Keywords: | H-eigenvalue Laplacian Loose cycle Loose path Regular uniform hypergraph Tight cycle Tight path |

Issue Date: | 2014 |

Publisher: | North-Holland |

Source: | Linear algebra and its applications, 2014, v. 443, p. 215-227 How to cite? |

Journal: | Linear algebra and its applications |

Abstract: | In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k≥3, reaches its upper bound 2Δ(G), where Δ(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and odd-bipartite. We show that an s-cycle G, as a k-uniform hypergraph, where 1≤s≤k-1, is regular if and only if there is a positive integer q such that k=q(k-s). We show that an even-uniform s-path and an even-uniform non-regular s-cycle are always odd-bipartite. We prove that a regular s-cycle G with k=q(k-s) is odd-bipartite if and only if m is a multiple of 2t0, where m is the number of edges in G, and q=2t0(2 l0+1) for some integers t0 and l0. We identify the value of the largest signless Laplacian H-eigenvalue of an s-cycle G in all possible cases. When G is odd-bipartite, this is also its largest Laplacian H-eigenvalue. We introduce supervertices for hypergraphs, and show the components of a Laplacian H-eigenvector of an odd-uniform hypergraph are equal if such components correspond vertices in the same supervertex, and the corresponding Laplacian H-eigenvalue is not equal to the degree of the supervertex. Using this property, we show that the largest Laplacian H-eigenvalue of an odd-uniform generalized loose s-cycle G is equal to Δ(G)=2. We also show that the largest Laplacian H-eigenvalue of a k-uniform tight s-cycle G is not less than Δ(G)+1, if the number of vertices is even and k=4l+3 for some nonnegative integer l. |

URI: | http://hdl.handle.net/10397/16260 |

ISSN: | 0024-3795 |

EISSN: | 1873-1856 |

DOI: | 10.1016/j.laa.2013.11.008 |

Appears in Collections: | Journal/Magazine Article |

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