Geodesic
Upper bounds for the second largest eigenvalue of symmetric nonnegative matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 138-163 
L. Yu. Kolotilina. Upper bounds for the second largest eigenvalue of symmetric nonnegative matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 138-163. http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a11/
@article{ZNSL_2012_405_a11,
     author = {L. Yu. Kolotilina},
     title = {Upper bounds for the second largest eigenvalue of symmetric nonnegative matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {138--163},
     year = {2012},
     volume = {405},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a11/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper suggests upper bounds on the second largest eigenvalue and the sum of two largest eigenvalues of symmetric nonnegative matrices and graphs. Conditions necessary and sufficient for some of the bounds to be attained are established. Special attention is paid to the subclass of matrices with zero diagonal entries and with off-diagonal entries not exceeding unity, which obviously contains the adjacency matrices of undirected graphs.

[1] L. Yu. Kolotilina, “Otsenki i neravenstva dlya perronovskogo kornya neotritsatelnoi matritsy”, Zap. nauchn. semin. POMI, 284, 2002, 77–122 | MR | Zbl

[2] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York etc., 1979 | MR | Zbl

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

[4] J. Ebrahimi, B. Mohar, V. Nikiforov, A. S. Ahmady, “On the sum of two largest eigenvalues of a symmetric matrix”, Linear Algebra Appl., 429 (2008), 2781–2787 | DOI | MR | Zbl

[5] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990 | MR | Zbl

[6] M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Inc., Boston, 1964 | MR | Zbl

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

[8] W. So, “Commutativity and spectra of Hermitian matrices”, Linear Algebra Appl., 212/213 (1994), 121–129 | DOI | MR | Zbl