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 K-2,K-s+1, then the largest eigenvalue q(G) of the signless Laplacian of G satisfies q(G) <= n+2s/2 + 1/2 root(n-2s)(2) + 8s, with equality holding if and only if G is a join of K-1 and an s-regular graph of order n-1. (C) 2016 Elsevier Inc. All rights reserved.