On the second largest eigenvalue of the signless Laplacian

Let $G$ be a graph of order $n,$ and let $q_1(G)\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge q_n(G)$ be the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian of $G\mathrm{.}$ In this paper we give a necessary and sufficient condition for the equality $q_k(G)=n-2,$ where $1<k\le n\mathrm{.}$ 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 $K_{1,\mathrm{.}\mathrm{.}\mathrm{.},1,2,\mathrm{.}\mathrm{.}\mathrm{.},2}$.