Let G be a non-bipartite graph which does not contain any odd cycle of length at most $2k+1$. In this paper, we determine the maximum Q-index of G if its order is fixed, and the corresponding extremal graph is uniquely characterized. Moreover, if the size of G is given, the maximum Q-index of G and the unique extremal graph are also proved.
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Lu et al. (2022) studied this question.