On the second largest eigenvalue of the signless Laplacian

Let G be a graph of order n  , and let q1(G)≥⋯≥qn(G)q1(G)≥⋯≥qn(G) be the eigenvalues of the Q-matrix of G, also known as the signless Laplacian of G  . We give a necessary and sufficient condition for the equality qk(G)=n-2qk(G)=n-2, where 1