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+1K2,s+1, then the largest eigenvalue q(G)q(G) of the signless Laplacian of G satisfiesq(G)≤n+2s2+12(n−2s)2+8s, with equality holding if and only if G   is a join of K1K1 and an s  -regular graph of order n−1n−1.