Taiwanese Journal of Mathematics

Some Properties of the Signless Laplacian and Normalized Laplacian Tensors of General Hypergraphs

Cunxiang Duan, Ligong Wang, and Xihe Li

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Abstract

In this paper, we obtain some properties of signless Laplacian eigenvalues of general hypergraphs. We give the upper and the lower bound of edge connectivity of general hypergraphs in terms of average degree, minimum degree, the rank and the number of vertices, or analytic connectivity $\alpha(G)$, respectively. We also give the upper bound of analytic connectivity $\alpha(G)$ of general hypergraphs in terms of the degrees of vertices. Finally, we obtain the bounds of the smallest $H^{+}$-eigenvalue of the normalized Laplacian sub-tensors of general hypergraphs.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 17 pages.

Dates
First available in Project Euclid: 2 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1562054420

Digital Object Identifier
doi:10.11650/tjm/190606

Subjects
Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 15A18: Eigenvalues, singular values, and eigenvectors 05C40: Connectivity

Keywords
signless Laplacian tensor normalized Laplacian tensor general hypergraph edge connectivity $H^{+}$-eigenvalue

Citation

Duan, Cunxiang; Wang, Ligong; Li, Xihe. Some Properties of the Signless Laplacian and Normalized Laplacian Tensors of General Hypergraphs. Taiwanese J. Math., advance publication, 2 July 2019. doi:10.11650/tjm/190606. https://projecteuclid.org/euclid.twjm/1562054420


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References

  • A. Banerjee, A. Char and B. Mondal, Spectra of general hypergraphs, Linear Algebra Appl. 518 (2017), 14–30.
  • A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012.
  • S. M. Cioabă, D. A. Gregory and V. Nikiforov, Extreme eigenvalues of nonregular graphs, J. Combin. Theory Ser. B 97 (2007), no. 3, 483–486.
  • J. Cooper and A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012), no. 9, 3268–3292.
  • D. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts 75, Cambridge University Press, Cambridge, 2010.
  • C. Duan, L. Wang and P. Xiao, The largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices), arXiv:1807.05955.
  • C. Duan, L. Wang, P. Xiao and X. Li, The (signless Laplacian) spectral radius (of subgraphs) of uniform hypergraphs, submitted, 2017.
  • S. Hu and L. Qi, The Laplacian of a uniform hypergraph, J. Comb. Optim. 29 (2015), no. 2, 331–366.
  • H. Li, J.-Y. Shao and L. Qi, The extremal spectral radii of $k$-uniform supertrees, J. Comb. Optim. 32 (2016), no. 3, 741–764.
  • H. Li, J. Zhou and C. Bu, Principal eigenvectors and spectral radii of uniform hypergraphs, Linear Algebra Appl. 544 (2018), 273–285.
  • W. Li, J. Cooper and A. Chang, Analytic connectivity of $k$-uniform hypergraphs, Linear Multilinear Algebra 65 (2017), no. 6, 1247–1259.
  • L.-H. Lim, Singular values and eigenvalues of tensors: A variational approach, Proceedings of the 1st IEEE international Workshop on Computational Advances in Multi-sensor Adaptive Processing, (2005), 129–132.
  • ––––, Eigenvalues of tensors and some very basic spectral hypergraph theory, Matrix Computations and Scientific Computing Seminar, April 16, 2008. http://www.stat.uchicago.edu/\~lekheng/work/mcsc2.
  • H. Lin, B. Mo, B. Zhou and W. Weng, Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs, Appl. Math. Comput. 285 (2016), 217–227.
  • L. Liu, L. Kang and X. Yuan, On the principal eigenvectors of uniform hypergraphs, Linear Algebra Appl. 511 (2016), 430–446.
  • L. Lu and S. Man, Connected hypergraphs with small spectral radius, Linear Algebra Appl. 509 (2016), 206–227.
  • C. Ouyang, L. Qi and X. Yuan, The first few unicyclic and bicyclic hypergraphs with largest spectral radii, Linear Algebra Appl. 527 (2017), 141–162.
  • L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005), no. 6, 1302–1324.
  • ––––, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl. 439 (2013), no. 1, 228–238.
  • ––––, $H^{+}$-eigenvalues of Laplacian and signless Laplacian tensors, Commun. Math. Sci. 12 (2014), no. 6, 1045–1064.
  • P. Xiao and L. Wang, The maximum spectral radius of uniform hypergraphs with given number of pendant edges, Linear Multilinear Algebra 67 (2019), no. 7, 1392–1403.
  • P. Xiao, L. Wang and Y. Du, The first two largest spectral radii of uniform supertrees with given diameter, Linear Algebra Appl. 536 (2018), 103–119.
  • P. Xiao, L. Wang and Y. Lu, The maximum spectral radii of uniform supertrees with given degree sequences, Linear Algebra Appl. 523 (2017), 33–45.
  • X. Yuan, J. Shao and H. Shan, Ordering of some uniform supertrees with larger spectral radii, Linear Algebra Appl. 495 (2016), 206–222.
  • J.-J. Yue, L.-P. Zhang, M. Lu and L.-Q. Qi, The adjacency and signless Laplacian spectra of cored hypergraphs and power hypergraphs, J. Oper. Res. Soc. China 5 (2017), no. 1, 27–43.
  • W. Zhang, L. Liu, L. Kang and Y. Bai, Some properties of the spectral radius for general hypergraphs, Linear Algebra Appl. 513 (2017), 103–119.