H-EIGENVALUES OF LAPLACIAN AND SIGNLESS LAPLACIAN TENSORS

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.

[1]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[2]  L. Qi.  Eigenvalues and invariants of tensors , 2007 .

[3]  Kung-Ching Chang,et al.  Perron-Frobenius theorem for nonnegative tensors , 2008 .

[4]  Marcello Pelillo,et al.  New Bounds on the Clique Number of Graphs Based on Spectral Hypergraph Theory , 2009, LION.

[5]  Michael K. Ng,et al.  Finding the Largest Eigenvalue of a Nonnegative Tensor , 2009, SIAM J. Matrix Anal. Appl..

[6]  Yongjun Liu,et al.  An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor , 2010, J. Comput. Appl. Math..

[7]  Luc T. Ikelle,et al.  Appendix B - Nonnegative Tensor Factorization , 2010 .

[8]  Qingzhi Yang,et al.  Further Results for Perron-Frobenius Theorem for Nonnegative Tensors , 2010, SIAM J. Matrix Anal. Appl..

[9]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[10]  Joshua N. Cooper,et al.  Spectra of Uniform Hypergraphs , 2011, 1106.4856.

[11]  Tan Zhang,et al.  Primitivity, the Convergence of the NQZ Method, and the Largest Eigenvalue for Nonnegative Tensors , 2011, SIAM Journal on Matrix Analysis and Applications.

[12]  Yi,et al.  LINEAR CONVERGENCE OF THE LZI ALGORITHM FOR WEAKLY POSITIVE TENSORS , 2012 .

[13]  L. Qi.  Symmetric nonnegative tensors and copositive tensors , 2012, 1211.5642.

[14]  Liqun Qi,et al.  Algebraic connectivity of an even uniform hypergraph , 2012, J. Comb. Optim..

[15]  Jinshan Xie,et al.  On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs , 2013 .

[16]  Jinshan Xie,et al.  H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph , 2013 .

[17]  S. Gaubert,et al.  Perron–Frobenius theorem for nonnegative multilinear forms and extensions , 2009, 0905.1626.

[18]  Jinshan Xie,et al.  On the Z‐eigenvalues of the signless Laplacian tensor for an even uniform hypergraph , 2013, Numer. Linear Algebra Appl..

[19]  L. Qi,et al.  Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues , 2013, 1304.6839.

[20]  L. Qi,et al.  The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph , 2013, 1304.1315.

[21]  Guoyin Li,et al.  The Z‐eigenvalues of a symmetric tensor and its application to spectral hypergraph theory , 2013, Numer. Linear Algebra Appl..

[22]  Yi Xu,et al.  Nonnegative Tensor Factorization, Completely Positive Tensors, and a Hierarchical Elimination Algorithm , 2013, SIAM J. Matrix Anal. Appl..

[23]  L. Qi,et al.  Strictly nonnegative tensors and nonnegative tensor partition , 2011, Science China Mathematics.

[24]  Liqun Qi,et al.  The eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a uniform hypergraph , 2013, Discret. Appl. Math..

[25]  Tan Zhang,et al.  On Spectral Hypergraph Theory of the Adjacency Tensor , 2012, Graphs Comb..

[26]  Liqun Qi,et al.  M-Tensors and Some Applications , 2014, SIAM J. Matrix Anal. Appl..

[27]  Liqun Qi,et al.  The Laplacian of a uniform hypergraph , 2015, J. Comb. Optim..

[28]  Marina Weber,et al.  Using Algebraic Geometry , 2016 .