Maxima Q-index of graphs with forbidden odd cycles

Let $q\left( G\right)$ be the $Q$-index (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 $n-k.$ The main result of this paper is the following theorem: Let $k\geq3,$ $n\geq110k^{2},$ and $G$ be a graph of order $n$. If $G$ has no $C_{2k+1},$ then $q\left( G\right) <q\left( S_{n,k}\right) ,$ unless $G=S_{n,k}.$ This result proves the odd case of the conjecture in [M.A.A. de Freitas, V. Nikiforov, and L. Patuzzi, Maxima of the $Q$-index: forbidden $4$-cycle and $5$-cycle, \emph{Electron. J. Linear Algebra }26 (2013), 905-916.]