Skip to main content
SpringerLink
Log in

H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brouwer A E, Haemers W H. Spectra of Graphs. Berlin: Springer, 2011

    Google Scholar 

  2. Cartwright D, Sturmfels B. The number of eigenvalues of a tensor. Linear Algebra Appl, 2013, 438(2): 942–952

    Article  MathSciNet  Google Scholar 

  3. Chang K C, Pearson K, Zhang T. Perron Frobenius Theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507–520

    MathSciNet  MATH  Google Scholar 

  4. Chang K C, Pearson K, Zhang T. Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. School of Mathematical Sciences, Peking University, Beijing, China, Preprint, 2010

    Google Scholar 

  5. Chang K C, Pearson K, Zhang T. Some variational principles of the Z-eigenvalues for nonnegative tensors. School of Mathematical Sciences, Peking University, Beijing, China, Preprint, 2011. http://www.mathinst.pku.edu.cn/data/upload/month_201112/2011-no44_dp47rB.pdf

    Google Scholar 

  6. Chang K C, Qi L, Zhou G. Singular values of a real rectangular tensor. J Math Anal Appl, 2010, 370: 284–294

    Article  MathSciNet  MATH  Google Scholar 

  7. Chung F R K. Spectral Graph Theory. Providence: Amer Math Soc, 1997

    MATH  Google Scholar 

  8. Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436(9): 3268–3292

    Article  MathSciNet  MATH  Google Scholar 

  9. Cvetković D, Rowlinson P, Simić S K. Eigenvalue bounds for the signless Laplacian. Publ Inst Math (Beograd), 2007, 95(81): 11–27

    Article  Google Scholar 

  10. Fiedler M. Algebraic connectivity of graphs. Czech Math J, 1973, 98(23): 298–305

    MathSciNet  Google Scholar 

  11. Hu S, Qi L. Algebraic connectivity of an even uniform hypergraph. J Comb Optim, 2012, 24(4): 564–579

    Article  MathSciNet  MATH  Google Scholar 

  12. Kolda T G, Bader B W. Tensor decompositions and applications. SIAM Review, 2009, 51: 455–500

    Article  MathSciNet  MATH  Google Scholar 

  13. Lim L -H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP ′05), Vol 1. 2005, 129–132

    Google Scholar 

  14. Ni G, Qi L, Wang F, Wang Y. The degree of the E-characteristic polynomial of an even order tensor. J Math Anal Appl, 2007, 329: 1218–1229

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl, 2007, 325: 1363–1377

    Article  MathSciNet  MATH  Google Scholar 

  17. Qi L. The spectral theory of tensors. arXiv: 1201.3424v1 [math.SP]

  18. Qi L, Sun W, Wang Y. Numerical multilinear algebra and its applications. Front Math China, 2007, 2(4): 501–526

    Article  MathSciNet  MATH  Google Scholar 

  19. Rota Bulò S, Pelillo M. A generalization of the Motzkin-Straus theorem to hypergraphs. Optim Lett, 2009, 3: 287–295

    Article  MathSciNet  MATH  Google Scholar 

  20. Rota Bulò S, Pelillo M. New bounds on the clique number of graphs based on spectral hypergraph theory. In: Stützle T, ed. Learning and Intelligent Optimization. Berlin: Springer-Verlag, 2009, 45–48

    Chapter  Google Scholar 

  21. Van Loan C. Future directions in tensor-based computation and modeling. Workshop Report in Arlington, Virginia at National Science Foundation, February 20–21, 2009. http://www.cs.cornell.edu/cv/TenWork/Home.htm

  22. Yang Q, Yang Y. Further results for Perron-Frobenius Theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32: 1236–1250

    Article  MathSciNet  MATH  Google Scholar 

  23. Yang Y, Yang Q. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517–2530

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang Y, Yang Q. Singular values of nonnegative rectangular tensors. Front Math China, 2011, 6(2): 363–378

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Discrete Mathematics, Fuzhou University, Fuzhou, 350003, China

    Jinshan Xie & An Chang

  2. School of Mathematics and Computer Science, Longyan University, Longyan, 364012, China

    Jinshan Xie

Authors
  1. Jinshan Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. An Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to An Chang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xie, J., Chang, A. H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph. Front. Math. China 8, 107–127 (2013). https://doi.org/10.1007/s11464-012-0266-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-012-0266-6

Keywords

MSC

Search

Navigation