Spectral directed hypergraph theory via tensors. Issue 4 (2nd April 2016)

• Record Type: Journal Article
• Authors: Xie, Jinshan; Qi, Liqun
• Is Part Of: Linear & multilinear algebra. Volume 64:Issue 4(2016)
• Page Start: 780
• Page End: 794
• Publication Date: 2016-04-02
• Subjects: Spectrum -- directed hypergraph -- H-eigenvalue -- adjacency tensor -- Laplacian tensor -- signless Laplacian tensor; 05C65 -- 15A18; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2015.1125838
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : In this paper, we show that each of the adjacency tensor, the Laplacian tensor and the signless Laplacian tensor of a uniform directed hypergraph has n linearly independent H -eigenvectors. Some lower and upper bounds for the largest and smallest adjacency, Laplacian and signless Laplacian H -eigenvalues of a uniform directed hypergraph are given. For a uniform directed hypergraph, the smallest Laplacian H -eigenvalue is 0. On the other hand, the upper bound of the largest adjacency and signless Laplacian H -eigenvalues are achieved if and only if it is a complete directed hypergraph. For a uniform directed hyperstar, all adjacency H -eigenvalues are 0. At the same time, we make some conjectures about the nonnegativity of one H -eigenvector corresponding to the largest H -eigenvalue, and raise some questions about whether the Laplacian and signless Laplacian tensors are positive semi-definite for a uniform directed hypergraph.