  Let G be a graph of order n and let q(G) be that largest eigenvalue of the
signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2,
then G contains cycles of length l whenever 2<l<2k+3. This bound is
asymptotically tight. It implies an asymptotic solution to a recent conjecture
about the maximum q(G) of a graph G with no cycle of a specified length.
