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Let $G$ be a non-bipartite graph which does not contain any odd cycle oflength 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 uniquelycharacterized. Moreover, if the size of $G$ is given, the maximum $Q$-index of$G$ and the unique extremal graph are also proved.

Maxima of the $Q$-index of non-bipartite graphs: forbidden short odd cycles