Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs

Abstract

We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest (signless) Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges (2d≥k≥3), we show that its largest (signless) Laplacian Z-eigenvalue is d.

Key words： Hypergraph eigenvalue; Laplacian tensor; signless Laplacian tensor; Z-eigenvalue

References

1 Bu C, Zhou J, Wei Y. E-cospectral hypergraphs and some hypergraphs determined by their spectra. Linear Algebra Appl, 2014, 459: 397–403

2 Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3292

3 Cooper J, Dutle A. Computing hypermatrix spectra with the Poisson product formula. Linear Multilinear Algebra, 2015, 63: 956–970

4 Cvetković D, Rowlinson P, Simić S. An Introduction to the Theory of Graph Spectra. Cambridge: Cambridge University Press, 2010

5 Hu S, Qi L. The eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a uniform hypergraph. Discrete Appl Math, 2014, 169: 140–151

6 Hu S, Qi L, Xie J. The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph. Linear Algebra Appl, 2015, 469: 1–27

7 Khan M, Fan Y. On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs. arXiv: 1408.3303

8 Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multisensor Adaptive Processing. 2005, 129–132

9 Pearson K, Zhang T. On spectral hypergraph theory of the adjacency tensor. Graphs Combin, 2014, 30: 1233–1248

10 Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324

11 Qi L. H⁺-eigenvalues of Laplacian and signless Laplacian tensors. Commun Math Sci, 2014, 12: 1045–1064

12 Shao J. A general product of tensors with applications. Linear Algebra Appl, 2013, 439: 2350–2366

13 Shao J, Qi L, Hu S. Some new trace formulas of tensors with applications in spectral hypergraph theory. Linear Multilinear Algebra, 2015, 63: 971–992

14 Shao J, Shan H, Wu B. Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs. Linear Multilinear Algebra, DOI: 10.1080/03081087.2015.1009061

15 Song Y, Qi L. Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J Matrix Anal Appl, 2013, 34: 1581–1595

16 Xie J, Chang A. On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs. Linear Algebra Appl, 2013, 439: 2195–2204

17 Zhou J, Sun L, Wang W, Bu C. Some spectral properties of uniform hypergraphs. Electron J Combin, 2014, 21: P4.24