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Graph Theory 62 (2009) 362--368] gave the spectral Stability Lemma, which implies that for every $\\varepsilon>0$, sufficiently large $n$ and a non-bipartite graph $H$ with chromatic number $\\chi(H)$, the extremal graph for ${\\rm spex}(n,H)$ can be obtained from the Tur\\'{a}n graph $T_{\\chi(H)-1}(n)$ by adding and deleting at most $\\varepsilon n^2$ edges. 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