Geodesic
Bounds for the extreme eigenvalues of the Laplacian and signless Laplacian of a graph
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 104-123 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper suggests a new approach to deriving lower bounds for the Laplacian spectral radius and upper bounds for the smallest eigenvalue of the signless Laplacian of an undirected simple $r$-partite graph on $n$ vertices, $2\le r\le n$. The approach is based on inequalities for the extreme eigenvalues of a block-partitioned Hermitian matrix, established earlier, and on the Rayleigh principle. Specific lower and upper bounds, generalizing and extending known results from $r=2$ to $r\ge2$ are considered, and the cases where these bounds are sharp are described. 
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     author = {L. Yu. Kolotilina},
     title = {Bounds for the extreme eigenvalues of the {Laplacian} and signless {Laplacian} of a~graph},
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L. Yu. Kolotilina. Bounds for the extreme eigenvalues of the Laplacian and signless Laplacian of a graph. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 104-123. http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a9/

[1] L. Yu. Kolotilina, “Ob uluchshenii otsenok Chistyakova dlya perronovskogo kornya neotritsatelnoi matritsy”, Zap. nauchn. semin. POMI, 346, 2007, 103–118 | MR

[2] L. Yu. Kolotilina, “Neravenstva dlya krainikh sobstvennykh znachenii blochnykh ermitovykh matrits s prilozheniyami k spektralnoi teorii grafov”, Zap. nauchn. semin. POMI, 382, 2010, 82–103 | MR

[3] L. Yu. Kolotilina, “Teorema Ostrovskogo o krugakh i nizhnie otsenki dlya naimenshikh sobstvennykh i singulyarnykh znachenii”, Zap. nauchn. semin. POMI, 382, 2010, 125–140 | MR

[4] L. Yu. Kolotilina, “Novye konturnye otsenki dlya perronovskogo kornya neotritsatelnoi matrits”, Zap. nauchn. semin. POMI, 395, 2011, 86–103

[5] Kinkar Ch. Das, “On conjectures involving second largest Laplacian eigenvalue of graphs”, Linear Algebra Appl., 432 (2010), 3018–3029 | DOI | MR | Zbl

[6] R. Grone, R. Merris, “The Laplacian spectrum of a graph II”, SIAM J. Discrete Math., 7 (1994), 221–229 | DOI | MR | Zbl

[7] Y. Hong, X. Zhang, “Sharp upper and lower bounds for the largest eigenvalue of the Laplacian matrices of trees”, Discrete Math., 296 (2005), 187–197 | DOI | MR | Zbl

[8] H. Minc, Nonnegative Matrices, John Wiley and Sons, New York etc., 1988 | MR

[9] V. Nikiforov, “Chromatic number and spectral radius”, Linear Algebra Appl., 426 (2007), 810–814 | DOI | MR | Zbl

[10] C. S. Oliveira, L. S. de Lima, N. M. M. de Abreu, P. Hansen, Bounds on the index of the signless Laplacian of a graph, Les Cahiers du GERAD, G-2007-72, 2008

[11] Gui-Xian Tian, Ting-Zhu Huang, Bo Zhou, “A note on sum of powers of the Laplacian eigenvalues of bipartite graphs”, Linear Algebra Appl., 430 (2009), 2503–2510 | DOI | MR | Zbl

[12] Aimei Yu, Mei Lu, Feng Tian, “On the spectral radius of graphs”, Linear Algebra Appl., 387 (2004), 41–49 | DOI | MR | Zbl

[13] Bo Zhou, “On sum of powers of the Laplacian eigenvalues of graphs”, Linear Algebra Appl., 429 (2008), 2239–2246 | DOI | MR | Zbl