  Let $G$ be a graph of order $n$ and let $q\left( G\right) $ be the largest
eigenvalue of the signless Laplacian of $G$. Let $S_{n,k}$ be the graph
obtained by joining each vertex of a complete graph of order $k$ to each vertex
of an independent set of order $n-k;$ and let $S_{n,k}^{+}$ be the graph
obtained by adding an edge to $S_{n,k}.$
  It is shown that if $k\geq2,$ $n\geq400k^{2},$ and $G$ is a graph of order
$n,$ with no cycle of length $2k+2,$ then $q\left( G\right) <q\left(
S_{n,k}^{+}\right) ,$ unless $G=S_{n,k}^{+}.$ This result completes the proof
of a conjecture of de Freitas, Nikiforov and Patuzzi.
