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Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 2, pp 583–591 | Cite as

Sharp Bounds for the Signless Laplacian Spectral Radius of Uniform Hypergraphs

  • Jun HeEmail author
  • Yan-Min Liu
  • Jun-Kang Tian
  • Xiang-Hu Liu
Original Paper
  • 56 Downloads

Abstract

Let \(\mathcal {H}\) be a k-uniform hypergraph on n vertices with degree sequence \(\Delta =d_1 \ge \cdots \ge d_n=\delta \). \(E_i\) denotes the set of edges of \(\mathcal {H}\) containing i. The average 2-degree of vertex i of \(\mathcal {H}\) is \(m_i = {\sum \nolimits _{\{ i,i_2 , \ldots i_k \} \in E_i } {d_{i_2 } \ldots d_{i_k } } } / d_i^{k - 1}\). In this paper, in terms of \(m_i\) and \(d_i\), we give some upper bounds and lower bounds for the spectral radius of the signless Laplacian tensor (\(Q(\mathcal {H})\)) of \(\mathcal {H}\). Some examples are given to show the tightness of these bounds.

Keywords

Hypergraph Adjacency tensor Signless Laplacian tensor Spectral radius 

Mathematics Subject Classification

15A42 05C50 

Notes

Acknowledgements

Jun He is supported by the Science and Technology Foundation of Guizhou Province (Qian ke he Ji Chu [2016]1161); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2016]255); the Doctoral Scientific Research Foundation of Zunyi Normal College (BS[2015]09); High-level Innovative Talents of Guizhou Province (Zun Ke He Ren Cai[2017]8). Yan-Min Liu is supported by National Science Foundations of China (71461027); Science and Technology Talent Training Object of Guizhou Province outstanding youth (Qian ke he ren zi [2015]06); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2014]295); 2013, 2014 and 2015 Zunyi 15851 Talents Elite Project funding; Zhunyi Innovative Talent Team (Zunyi KH(2015)38); Innovative talent team in Guizhou Province (Qian Ke HE Pingtai Rencai[2016]5619). Tian is supported by Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2015]451); Science and Technology Foundation of Guizhou Province (Qian ke he J zi [2015]2147). Xiang-Hu Liu is supported by the Guizhou Province Department of Education fund (KY[2015]391, [2016]046); Guizhou Province Department of Education Teaching Reform Project [2015]337; Guizhou Province Science and Technology fund (Qian Ke He Ji Chu[2016]1160).

References

  1. 1.
    Cooper, J., Dutle, A.: Spectra of uniform hypergraphs. Linear Algebra Appl. 436, 3268–3292 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Pearson, K., Zhang, T.: On spectral hypergraph theory of the adjacency tensor. Graphs Combin. 30, 1233–1248 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symbol Comput. 40, 1302–1324 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Qi, L.: \(H^+\)-eigenvalues of Laplacian and signless Laplacian tensors. Commun. Math. Sci. 12, 1045–1064 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Shao, J.-Y.: A general product of tensors with applications. Linear Algebra Appl. 439, 2350–2366 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Yang, Y., Yang, Q.: Further results for Perron-Frobenius theorem for nonegative tensors. SIAM J. Matrix Anal. Appl. 31, 2517–2530 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Yuan, X., Zhang, M., Lu, M.: Some upper bounds on the eigenvalues of uniform hypergraphs. Linear Algebra Appl. 484, 540–549 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lin, H., Mo, B., Zhou, B., Weng, W.: Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs. Appl. Math. Comput. 285, 217–227 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Khan, M., Fan, Y.Z.: On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs. Linear Algebra Appl. 480, 93–106 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, C., Chen, Z., Li, Y.: A new eigenvalue inclusion set for tensors and its applications. Linear Algebra Appl. 481, 36–53 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Friedland, S., Gaubert, A., Han, L.: Perron–Frobenius theorems for nonnegative multilinear forms and extensions. Linear Algebra Appl. 438, 738–749 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yang, Y., Yang, Q.: On some properties of nonegative weakly irreducible tensors (2011). Preprint available at arXiv:1111.0713v2

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • Jun He
    • 1
    Email author
  • Yan-Min Liu
    • 1
  • Jun-Kang Tian
    • 1
  • Xiang-Hu Liu
    • 1
  1. 1.School of mathematicsZunyi Normal CollegeZunyiPeople’s Republic of China

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