Graphs with the second signless Laplacian eigenvalue <= 4

We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is <= 4. We discover that the question is inextricable linked to a knap-sack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.