Tur\'an numbers for non-bipartite graphs and applications to spectral extremal problems

Given a graph family H with minH∈ Hχ(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(Tn,r)+ n/2r. 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 Gy{o}ri, who settled the case that F₁=⋯ =Fₖ = Kᵣ₊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(Tn,r)+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(Tn,r)+ n/2r. 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.