Maxima of the Q-index: Forbidden even cycles

Let G be a graph of order n   and let q(G)q(G) be the largest eigenvalue of the signless Laplacian of G  . Let Sn,kSn,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−kn−k; let Sn,k+ be the graph obtained by adding an edge to Sn,kSn,k.It is shown that if k≥2k≥2, n≥400k2n≥400k2, and G is a graph of order n  , with no cycle of length 2k+22k+2, then q(G)