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 κi(G) be the ith largest eigenvalue of the signless Laplacian of G, we show that κi+1(G)-1⩽κi(G-v)⩽κi(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.