Maxima of the $Q$-index of non-bipartite $C_3$-free graphs

A classic result in extremal graph theory, known as Mantel’s theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^2}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel’s theorem for given order $n\mathrm{.}$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size $m\mathrm{.}$ In this paper, we prove $Q$-spectral versions of Mantel’s theorem.