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

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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. 
@article{ZNSL_2011_395_a9,
     author = {L. Yu. Kolotilina},
     title = {Bounds for the extreme eigenvalues of the {Laplacian} and signless {Laplacian} of a~graph},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {104--123},
     publisher = {mathdoc},
     volume = {395},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a9/}
}
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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/