A sharp upper bound for the spectral radius of a nonnegative matrix and applications

Lihua You; Yujie Shu; Xiao-Dong Zhang

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 701-715
  • ISSN: 0011-4642

Abstract

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We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results.

How to cite

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You, Lihua, Shu, Yujie, and Zhang, Xiao-Dong. "A sharp upper bound for the spectral radius of a nonnegative matrix and applications." Czechoslovak Mathematical Journal 66.3 (2016): 701-715. <http://eudml.org/doc/286837>.

@article{You2016,
abstract = {We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results.},
author = {You, Lihua, Shu, Yujie, Zhang, Xiao-Dong},
journal = {Czechoslovak Mathematical Journal},
keywords = {nonnegative matrix; spectral radius; graph; digraph},
language = {eng},
number = {3},
pages = {701-715},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A sharp upper bound for the spectral radius of a nonnegative matrix and applications},
url = {http://eudml.org/doc/286837},
volume = {66},
year = {2016},
}

TY - JOUR
AU - You, Lihua
AU - Shu, Yujie
AU - Zhang, Xiao-Dong
TI - A sharp upper bound for the spectral radius of a nonnegative matrix and applications
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 701
EP - 715
AB - We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results.
LA - eng
KW - nonnegative matrix; spectral radius; graph; digraph
UR - http://eudml.org/doc/286837
ER -

References

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  1. Aouchiche, M., Hansen, P., Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013), 21-33. (2013) Zbl1282.05086MR3045220
  2. Berman, A., Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics New York, Academic Press (1979). (1979) Zbl0484.15016MR0544666
  3. Bozkurt, S. B., Bozkurt, D., On the signless Laplacian spectral radius of digraphs, Ars Comb. 108 (2013), 193-200. (2013) Zbl1289.05270MR3060265
  4. Cao, D., Bounds on eigenvalues and chromatic numbers, Linear Algebra Appl. 270 (1998), 1-13. (1998) Zbl0894.05041MR1484072
  5. Chen, Y.-H., Pan, R.-Y., Zhang, X.-D., 10.1142/S1793830911001152, Discrete Math. Algorithms Appl. 3 (2011), 185-192. (2011) Zbl1222.05149MR2822283DOI10.1142/S1793830911001152
  6. Cui, S.-Y., Tian, G.-X., Guo, J.-J., A sharp upper bound on the signless Laplacian spectral radius of graphs, Linear Algebra Appl. 439 (2013), 2442-2447. (2013) Zbl1282.05072MR3091317
  7. Das, K. Ch., A characterization on graphs which achieve the upper bound for the largest Laplacian eigenvalue of graphs, Linear Algebra Appl. 376 (2004), 173-186. (2004) Zbl1042.05059MR2015532
  8. Das, K. Ch., Kumar, P., 10.1016/j.disc.2003.08.005, Discrete Math. 281 (2004), 149-161. (2004) Zbl1042.05060MR2047763DOI10.1016/j.disc.2003.08.005
  9. Fiedler, M., A geometric approach to the Laplacian matrix of a graph, Combinatorial and Graph-Theoretical Problems in Linear Algebra R. A. Brualdi 73-98 Proc. Conf. Minnesota 1991. IMA Vol. Math. Appl. 50 Springer, New York (1993). (1993) Zbl0791.05073MR1240957
  10. Fiedler, M., Algebraic connectivity of graphs, Czech. Math. J. 23 (1973), 298-305. (1973) Zbl0265.05119MR0318007
  11. Guo, J.-M., 10.1016/j.laa.2004.10.022, Linear Algebra Appl. 400 (2005), 61-66. (2005) Zbl1062.05091MR2131916DOI10.1016/j.laa.2004.10.022
  12. Guo, J.-M., Li, J., Shiu, W. C., A note on the upper bounds for the Laplacian spectral radius of graphs, Linear Algebra Appl. 439 (2013), 1657-1661. (2013) Zbl1282.05117MR3073893
  13. Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge (2013). (2013) Zbl1267.15001MR2978290
  14. Li, J.-S., Pan, Y.-L., 10.1007/s10114-004-0332-4, Acta Math. Sin., Engl. Ser. 20 (2004), 803-806. (2004) Zbl1074.15026MR2121091DOI10.1007/s10114-004-0332-4
  15. Li, J.-S., Pan, Y.-L., de Caen's inequality and bounds on the largest Laplacian eigenvalue of a graph, Linear Algebra Appl. 328 (2001), 153-160. (2001) Zbl0988.05062MR1823515
  16. Li, J.-S., Zhang, X.-D., On the Laplacian eigenvalues of a graph, Linear Algebra Appl. 285 (1998), 305-307. (1998) Zbl0931.05052MR1653547
  17. Oliveira, C. S., Lima, L. S. de, Abreu, N. M. M. de, Hansen, P., 10.1016/j.dam.2009.06.023, Discrete Appl. Math. 158 (2010), 355-360. (2010) Zbl1225.05174MR2588119DOI10.1016/j.dam.2009.06.023
  18. Shu, J., Wu, Y., 10.1016/j.laa.2003.07.015, Linear Algebra Appl. 377 (2004), 241-248. (2004) Zbl1030.05073MR2021614DOI10.1016/j.laa.2003.07.015
  19. Zhang, X.-D., Two sharp upper bounds for the Laplacian eigenvalues, Linear Algebra Appl. 376 (2004), 207-213. (2004) Zbl1037.05032MR2015534
  20. Zhang, X.-D., Li, J.-S., 10.1007/s101140200157, Acta Math. Sin., Engl. Ser. 18 (2002), 293-300. (2002) Zbl1019.15007MR1910965DOI10.1007/s101140200157
  21. Zhang, X.-D., Luo, R., 10.1016/S0024-3795(02)00509-8, Linear Algebra Appl. 362 (2003), 109-119. (2003) Zbl1017.05078MR1955457DOI10.1016/S0024-3795(02)00509-8

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