Let G be a graph of order n, and let q1(G)≥⋯≥q n(G) be the eigenvalues of the Q-matrix of G, also known as the signless Laplacian of G. We give a necessary and sufficient condition for the equality qk(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 thatq2(G)≥δ(G)and determine all graphs for which equality holds. © 2012 Elsevier Inc. All rights reserved.
De Lima, L. S., & Nikiforov, V. (2013). On the second largest eigenvalue of the signless Laplacian. Linear Algebra and Its Applications, 438(3), 1215–1222. https://doi.org/10.1016/j.laa.2012.07.052