On the second largest eigenvalue of the signless Laplacian

Let G be a graph of order n, and let q(1) (G) >= ... >= q(n) (G) be the eigenvalues of the Q-matrix of C. also known as the signless Laplacian of G. We give a necessary and sufficient condition for the equality q(k) (G) = n - 2, where 1 < k <= n. In particular, this result solves an open problem raised by Wang, Belardo, Huang and Borovicanin. We also show that q(2) (G) >= delta (G) and determine all graphs for which equality holds. (C) 2012 Elsevier Inc. All rights reserved.