A sharp spectral extremal result for general non-bipartite graphs

For a graph family $\mathcal{F}$, let $\mathrm{e}\mathrm{x}(n,\mathcal{F})$ and $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{x}(n,\mathcal{F})$ denote the maximum number of edges and maximum spectral radius of an $n$-vertex $\mathcal{F}$-free graph, respectively, and let $\mathrm{E}\mathrm{X}(n,\mathcal{F})$ and $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,\mathcal{F})$ denote the corresponding sets of extremal graphs. Wang, Kang, and Xue showed that if $r\ge 2$ and $\mathrm{e}\mathrm{x}(n,F)=e(T_{n,r})+O(1)$ then $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,\mathcal{F})\subseteq \mathrm{E}\mathrm{X}(n,\mathcal{F})$ for $n$ large enough. Fang, Tait, and Zhai extended this result by showing if $e(T_{n,r})\le \mathrm{e}\mathrm{x}(n,\mathcal{F})<e(T_{n,r})+\lfloor n\mathrm{/}2r\rfloor$ then $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,\mathcal{F})\subseteq \mathrm{E}\mathrm{X}(n,\mathcal{F})$ for $n$ large enough, and asked for the maximum constant $c(r)$ such that $\mathrm{e}\mathrm{x}(n,\mathcal{F})\le e(T_{n,r})+(c(r)-\epsilon )n$ guarantees such containment. In this paper we determine $c(r)$ exactly for all $r\ge 3$.