Maxima of the Q-index: graphs with no K_s,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 $K_{2,s+1},$ then the largest eigenvalue $q(G)$ of the signless Laplacian of $G$ satisfies $$q(G)\leq\frac{n+2s}{2}+\frac{1}{2}\sqrt{(n-2s)^{2}+8s},$$ with equality holding if and only if $G$ is a join of $K_{1}$ and an $s$-regular graph of order $n-1.$