Maxima of the Q-index: forbidden odd cycles

Let q(G)q(G) be the Q-index (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. The main result of this paper is the following theorem:Let k≥3k≥3, n≥110k2n≥110k2, and G be a graph of order n. If G   has no C2k+1C2k+1, then q(G)