Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs

Kinkar Ch. Das; Muhuo Liu

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 1039-1048
  • ISSN: 0011-4642

Abstract

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In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with n vertices and clique number ω ( 2 ≤ ω ≤ n ) are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.

How to cite

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Das, Kinkar Ch., and Liu, Muhuo. "Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs." Czechoslovak Mathematical Journal 66.3 (2016): 1039-1048. <http://eudml.org/doc/286824>.

@article{Das2016,
abstract = {In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $\omega $$(2\le \omega \le n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.},
author = {Das, Kinkar Ch., Liu, Muhuo},
journal = {Czechoslovak Mathematical Journal},
keywords = {spectral radius; (signless) Laplacian spectral radius; clique number},
language = {eng},
number = {3},
pages = {1039-1048},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs},
url = {http://eudml.org/doc/286824},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Das, Kinkar Ch.
AU - Liu, Muhuo
TI - Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 1039
EP - 1048
AB - In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $\omega $$(2\le \omega \le n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.
LA - eng
KW - spectral radius; (signless) Laplacian spectral radius; clique number
UR - http://eudml.org/doc/286824
ER -

References

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  1. Aouchiche, M., Comparaison Automatise d'Invariants en Théorie des Graphes, PhD Thesis. École Polytechnique de Montréal (2006), French. (2006) MR2709332
  2. Bruardi, R. A., Hoffman, A. J., 10.1016/0024-3795(85)90092-8, Linear Algebra Appl. 65 (1985), 133-146. (1985) MR0774347DOI10.1016/0024-3795(85)90092-8
  3. Cai, G.-X., Fan, Y.-Z., The signless Laplacian spectral radius of graphs with given chromatic number, Math. Appl. 22 (2009), 161-167. (2009) Zbl1199.05210MR2847662
  4. Feng, L., Li, Q., Zhang, X.-D., 10.1016/j.aml.2005.11.030, Appl. Math. Lett. 20 (2007), 158-162. (2007) Zbl1109.05069MR2283904DOI10.1016/j.aml.2005.11.030
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  6. Hansen, P., Lucas, C., 10.1016/j.laa.2010.01.027, Linear Algebra Appl. 432 (2010), 3319-3336. (2010) Zbl1214.05079MR2639286DOI10.1016/j.laa.2010.01.027
  7. Hansen, P., Lucas, C., An inequality for the signless Laplacian index of a graph using the chromatic number, Graph Theory Notes N. Y. 57 (2009), 39-42. (2009) MR2666279
  8. He, B., Jin, Y.-L., Zhang, X.-D., Sharp bounds for the signless Laplacian spectral radius in terms of clique number, Linear Algebra Appl. 438 (2013), 3851-3861. (2013) Zbl1282.05119MR3034503
  9. Hoffman, A. J., Smith, J. H., On the spectral radii of topologically equivalent graphs, Recent Adv. Graph Theory, Proc. Symp. Prague 1974 Academia, Praha (1975), 273-281. (1975) Zbl0327.05125MR0404028
  10. Liu, J., Liu, B., 10.1007/s10587-008-0082-z, Czech. Math. J. 58 (2008), 1233-1240. (2008) Zbl1174.05079MR2471179DOI10.1007/s10587-008-0082-z
  11. Liu, M., Tan, X., Liu, B., 10.1007/s10587-010-0053-z, Czech. Math. J. 60 (2010), 849-867. (2010) Zbl1224.05311MR2672419DOI10.1007/s10587-010-0053-z
  12. Merris, R., Laplacian matrices of graphs: A survey, Linear Algebra Appl. 197-198 (1994), 143-176. (1994) Zbl0802.05053MR1275613
  13. Nikiforov, V., Bounds on graph eigenvalues II, Linear Algebra Appl. 427 (2007), 183-189. (2007) Zbl1128.05035MR2351351
  14. Stevanović, D., Hansen, P., The minimum spectral radius of graphs with a given clique number, Electron. J. Linear Algebra (electronic only) 17 (2008), 110-117. (2008) Zbl1148.05306MR2390363
  15. Zhang, J.-M., Huang, T.-Z., Guo, J.-M., The smallest signless Laplacian spectral radius of graphs with a given clique number, Linear Algebra Appl. 439 (2013), 2562-2576. (2013) Zbl1282.05164MR3095669

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