  Let $G$ be a graph of order $n,$ and let $q_{1}(G) \geq ...\geq q_{n}(G) $ be
the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian
of $G.$ In this paper we give a necessary and sufficient condition for the
equality $q_{k}(G) =n-2,$ where $1<k\leq n.$ In particular, this result solves
an open problem raised by Wang, Belardo, Huang and Borovicanin.
  We also show that [ q_{2}(G) \geq\delta(G)] and determine that equality holds
if and only if $G$ is one of the following graphs: a star, a complete regular
multipartite graph, the graph $K_{1,3,3},$ or a complete multipartite graph of
the type $K_{1,...,1,2,...,2}$.
