Abstract 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 κ 3 ( G ) and we give a lower bound in terms of the third largest degree d 3 of the graph G. In particular, we prove that κ 3 ( G ) ⩾ d 3 ( G ) - 2 . Furthermore, we show that in several situations the latter bound can be increased to d 3 - 1 .