Turán numbers for non-bipartite graphs and applications to spectral extremal problems

Given a graph family ℋ with min_H∈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)More