Maxima of the Q-index: Graphs with no Ks,t

This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order n that does not contain a specified complete bipartite subgraph. A conjecture is stated about general complete bipartite graphs, which is proved for infinitely many cases. More precisely, it is shown that if G is a graph of order n, with no subgraph isomorphic to K2,s+1, then the largest eigenvalue q(G) of the signless Laplacian of G satisfiesq(G) (Formula presented.), with equality holding if and only if G is a join of K1 and an s-regular graph of order n-1