Maxima Q-index of graphs with forbidden odd cycles

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Let q(G) be the Q-index (the largest eigenvalue of the signless Laplacian) of G. Let Sₙ,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≥3, n≥110k², and G be a graph of order n. If G has no C₂ₖ₊₁, then q(G) <q(Sₙ,ₖ), unless G=Sₙ,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, Electron. J. Linear Algebra 26 (2013), 905-916.]