In the paper, a new upper bound for the largest eigenvalue q1 of the signless Laplacian QG = DG + AG of a graph G, generalizing and improving the known bound q1 ≤ Δ1 + Δ2, where Δ1 ≥ ・・・ ≥ Δn are the ordered vertex degrees, and also new lower bounds for the second largest eigenvalue q2 of QG are proved. As implications, upper bounds for the difference q1 − μ1 of the largest eigenvalues of QG and of the Laplacian matrix LG = DG − AG, an upper bound for the largest eigenvalue of the adjacency matrix AG, and an upper bound for the difference q1 − q2 are obtained. All the bounds suggested are expressed in terms of the vertex degrees.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 419, 2013, pp. 139–153.
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Kolotilina, L.Y. Bounds for the Largest Two Eigenvalues of the Signless Laplacian. J Math Sci 199, 448–455 (2014). https://doi.org/10.1007/s10958-014-1872-5
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DOI: https://doi.org/10.1007/s10958-014-1872-5