The clique number and the smallest Q-eigenvalue of graphs

Let q_ (G) stand for the smallest eigenvalue of the signless Laplacian of a graph G of order n. This paper gives some results on the following extremal problem: How large can q_ (G) be if G is a graph of order n, with no complete subgraph of order r+1? It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds on q_ are obtained, thus extending previous work of Brandt for regular graphs. In addition, using graph blowups, a general asymptotic result about the maximum q_ is established. As a supporting tool, the spectra of the Laplacian and the signless Laplacian of blowups of graphs are calculated.