A general theorem in spectral extremal graph theory

The extremal graphs E X ( n , F ) \mathrm {EX}(n,\mathcal F) and spectral extremal graphs S P E X ( n , F ) \mathrm {SPEX}(n,\mathcal F) are the sets of graphs on n n vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in F \mathcal F . We prove a general theorem which allows us to characterize the spectral extremal graphs for a wide range of forbidden families F \mathcal F and implies several new and existing results. In particular, whenever E X ( n , F ) \mathrm {EX}(n,\mathcal F) contains the complete bipartite graph K k , n − k K_{k,n-k} (or certain similar graphs) then S P E X ( n , F ) \mathrm {SPEX}(n,\mathcal F) contains the same graph when n n is sufficiently large. We prove a similar theorem which relates S P E X <mml:mo stretchy="fal