| Title:
|
Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs (English) |
| Author:
|
Das, Kinkar Ch. |
| Author:
|
Liu, Muhuo |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
1039-1048 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $ $(2\leq \omega \leq n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved. (English) |
| Keyword:
|
spectral radius |
| Keyword:
|
(signless) Laplacian spectral radius |
| Keyword:
|
clique number |
| MSC:
|
05C50 |
| MSC:
|
05C75 |
| idZBL:
|
Zbl 06644049 |
| idMR:
|
MR3556883 |
| DOI:
|
10.1007/s10587-016-0308-4 |
| . |
| Date available:
|
2016-10-01T15:47:04Z |
| Last updated:
|
2023-10-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145887 |
| . |
| Reference:
|
[1] Aouchiche, M.: Comparaison Automatise d'Invariants en Théorie des Graphes.PhD Thesis. École Polytechnique de Montréal (2006), French. MR 2709332 |
| Reference:
|
[2] Bruardi, R. A., Hoffman, A. J.: On the spectral radius of $(0,1)$-matrices.Linear Algebra Appl. 65 (1985), 133-146. MR 0774347, 10.1016/0024-3795(85)90092-8 |
| Reference:
|
[3] Cai, G.-X., Fan, Y.-Z.: The signless Laplacian spectral radius of graphs with given chromatic number.Math. Appl. 22 (2009), 161-167. Zbl 1199.05210, MR 2847662 |
| Reference:
|
[4] Feng, L., Li, Q., Zhang, X.-D.: Spectral radii of graphs with given chromatic number.Appl. Math. Lett. 20 (2007), 158-162. Zbl 1109.05069, MR 2283904, 10.1016/j.aml.2005.11.030 |
| Reference:
|
[5] Guo, J.-M., Li, J., Shiu, W. C.: The smallest Laplacian spectral radius of graphs with a given clique number.Linear Algebra Appl. 437 (2012), 1109-1122. Zbl 1244.05143, MR 2926159 |
| Reference:
|
[6] Hansen, P., Lucas, C.: Bounds and conjectures for the signless Laplacian index of graphs.Linear Algebra Appl. 432 (2010), 3319-3336. Zbl 1214.05079, MR 2639286, 10.1016/j.laa.2010.01.027 |
| Reference:
|
[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. MR 2666279 |
| Reference:
|
[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. Zbl 1282.05119, MR 3034503 |
| Reference:
|
[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. Zbl 0327.05125, MR 0404028 |
| Reference:
|
[10] Liu, J., Liu, B.: The maximum clique and the signless Laplacian eigenvalues.Czech. Math. J. 58 (2008), 1233-1240. Zbl 1174.05079, MR 2471179, 10.1007/s10587-008-0082-z |
| Reference:
|
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| Reference:
|
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| Reference:
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[13] Nikiforov, V.: Bounds on graph eigenvalues II.Linear Algebra Appl. 427 (2007), 183-189. Zbl 1128.05035, MR 2351351 |
| Reference:
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[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. Zbl 1148.05306, MR 2390363 |
| Reference:
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[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. Zbl 1282.05164, MR 3095669 |
| . |
