Bounds for the Largest Two Eigenvalues of the Signless Laplacian.

In the paper, a new upper bound for the largest eigenvalue q1 of the signless Laplacian Q = D + A of a graph G, generalizing and improving the known bound q ≤ Δ + Δ, where Δ ≥ ・・・ ≥ Δ are the ordered vertex degrees, and also new lower bounds for the second largest eigenvalue q of Q are proved. As implications, upper bounds for the difference q − μ of the largest eigenvalues of Q and of the Laplacian matrix L = D − A, an upper bound for the largest eigenvalue of the adjacency matrix AG, and an upper bound for the difference q − q are obtained. All the bounds suggested are expressed in terms of the vertex degrees. [ABSTRACT FROM AUTHOR]