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Given a graph family H with ₇ ₇ (H) =r+1 3. Let ex (n, H) and spex (n, H) be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all H-free graphs of order n, respectively. Denote by EX (n, H) (resp. SPEX (n, H) ) the set of extremal graphs with respect to ex (n, H) (resp. spex (n, H) ). In this paper, we use a decomposition family defined by Simonovits to give a characterization of which graph families H satisfy ex (n, H) <e (T₍, ₑ) + n2r. Furthermore, we completely determine EX (n, G (F₁, , Fₖ) ) for n sufficiently large, where G (F₁, , Fₖ) denotes a finite graph family which consists of k edge-disjoint (r+1) -chromatic color-critical graphs F₁, , Fₖ. This result strengthens a theorem of Gyori, who settled the case that F₁= =Fₖ = Kₑ+₁. Wang, Kang and Xue %J. Combin. Theory Ser. B 159 (2023) 20--41 proved that SPEX (n, H) EX (n, H) for n sufficiently large and any graph H with ex (n, H) =e (T₍, ₑ) +O (1). As an application of our first theorem, we show that SPEX (n, H) EX (n, H) for n sufficiently large and any finite family H with ex (n, H) <e (T₍, ₑ) + n2r. As an application of our second theorem we completely determine SPEX (n, G (F₁, , Fₖ) ) for n sufficiently large. Finally, related problems are proposed for further research.
Fang et al. (Sat,) studied this question.