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Frontiers of Mathematics in China

, Volume 13, Issue 6, pp 1489–1499 | Cite as

Spectral radius of r-uniform supertrees with perfect matchings

  • Lei Zhang
  • An Chang
Research Article
  • 8 Downloads

Abstract

supertree is a connected and acyclic hypergraph. The set of r-uniform supertrees with n vertices and the set of r-uniform supertrees with perfect matchings on rk vertices are denoted by Tn and Tr,k, respectively. H. Li, J. Shao, and L. Qi [J. Comb. Optim., 2016, 32(3): 741–764] proved that the hyperstar Sn,r attains uniquely the maximum spectral radius in Tn. Focusing on the spectral radius in Tr,k, this paper will give the maximum value in Tr,k and their corresponding supertree.

Keywords

Supertrees spectral radius perfect matching 

MSC

15A42 05C50 05C65 

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Notes

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 11471077).

References

  1. 1.
    Berge C. Hypergraph: Combinatorics of Finite Sets. 3rd ed. Amsterdam: North-Holland, 1973Google Scholar
  2. 2.
    Bretto A. Hypergraph Theory: An Introduction. Berlin: Springer, 2013CrossRefzbMATHGoogle Scholar
  3. 3.
    Chang A. On the largest eigenvalue of a tree with perfect matchings. Discrete Math, 2003, 269: 45–63MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3292MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cvetković D, Doob M, Sachs H. Spectra of GraphTheory and Applications. New York: Academic Press, 1980zbMATHGoogle Scholar
  6. 6.
    Guo J M, Tan SW. On the spectral radius of trees. Linear Algebra Appl, 2001, 329: 1–8MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu S, Qi L, Shao J. Cored hypergraphs, power hypergraphs and their Laplacian eigen-values. Linear Algebra Appl, 2013, 439: 2980–2998MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li H, Shao J, Qi L. The extremal spectral radii of k-uniform supertrees. J Comb Optim, 2016, 32(3): 741–764MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lin H, Mo B, Zhou B, Weng W. Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs. Appl Math Comput, 2016, 285: 217–227MathSciNetGoogle Scholar
  10. 10.
    Lin H, Zhou B, Mo B. Upper bounds for H-and Z-spectral radii of uniform hyper-graphs. Linear Algebra Appl, 2016, 510: 205–221MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Qi L. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439: 228–238MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Schwenk A J, Wilson R J. Eigenvalues of graphs. In: Beineke L W, Wilson R J, eds. Selected Topics in Graph Theory. New York: Academic Press, 1978Google Scholar
  14. 14.
    Xiao P, Wang L, Lu Y. The maximum spectral radii of uniform supertrees with given degree sequences. Linear Algebra Appl, 2017, 523: 33–45MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Xu G H. On the spectral radius of trees with perfect matchings. In: Combinatorics and Graph Theory. Singapore: World Scientific, 1997Google Scholar
  16. 16.
    Yuan X. Ordering uniform supertrees by their spectral radii. Front Math China, 2017, 12(6): 1–16MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yuan X, Shao J, Shan H. Ordering of some uniform supertrees with larger spectral radii. Linear Algebra Appl, 2016, 495: 206–222MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yuan X, Zhang M, Lu M. Some upper bounds on the eigenvalues of uniform hyper-graphs. Linear Algebra Appl, 2015, 484: 540–549MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhang W, Kang L, Shan E, Bai Y. The spectra of uniform hypertrees. Linear Algebra Appl, 2017, 533: 84–94MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhou J, Sun L, Wang W, Bu C. Some spectral properties of uniform hypergraphs, Electron J Combin, 2014, 21(4): 4–24MathSciNetzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Discrete Mathematics and Theoretical Computer ScienceFuzhou UniversityFuzhouChina

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