A note on the signless Laplacian eigenvalues of graphs

In this paper, we consider the signless Laplacians of simple graphs and we give some eigenvalue inequalities. We first consider an interlacing theorem when a vertex is deleted. In particular, let G-v be a graph obtained from graph G by deleting its vertex v and κ_i(G) be the ith largest eigenvalue of the signless Laplacian of G, we show that κ_i+1(G)-1?κ_i(G-v)?κ_i(G). Next, we consider the third largest eigenvalue κ₃(G) and we give a lower bound in terms of the third largest degree d₃ of the graph G. In particular, we prove that . Furthermore, we show that in several situations the latter bound can be increased to d₃-1.