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

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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. 
@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},
     publisher = {mathdoc},
     volume = {405},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a11/}
}
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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/