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 $K_{ 1,...,1,2,...,2}$.
Categories: math.SP math.CO