Geodesic
Bounds for the largest two eigenvalues of the signless Laplacian
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVI, Tome 419 (2013), pp. 139-153 
L. Yu. Kolotilina. Bounds for the largest two eigenvalues of the signless Laplacian. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVI, Tome 419 (2013), pp. 139-153. http://geodesic.mathdoc.fr/item/ZNSL_2013_419_a8/
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     author = {L. Yu. Kolotilina},
     title = {Bounds for the largest two eigenvalues of the signless {Laplacian}},
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     year = {2013},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_419_a8/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In the paper, a new upper bound for the largest eigenvalue $q_1$ of the signless Laplacian $Q_G=D_G+A_G$ of a graph $G$, generalizing and improving the known bound $q_1\le\Delta_1+\Delta_2$, where $\Delta_1\ge\cdots\ge\Delta_n$ are the ordered vertex degrees, and new lower bounds for the second largest eigenvalue $q_2$ of $Q_G$ are proved. As implications, an upper bound for the difference $q_1-\mu_1$ of the largest eigenvalues of the signless Laplacian $Q_G$ and of the Laplacian $L_G=D_G-A_G$, an upper bound for the largest eigenvalue of the adjacency matrix $A_G$, and an upper bound for the difference $q_1-q_2$ are obtained. All the bounds suggested are expressed in terms of the vertex degrees.

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