An edge-spectral supersaturation of Mubayi's theorem for color-critical graphs

We study the supersaturation problem in its edge-spectral form. Let $\lambda(G)$ be the adjacency spectral radius of $G$. Nikiforov proved that every $K_{r+1}$-free graph $G$ with $m$ edges satisfies $\lambda (G)\le \sqrt{(1\!-\!1/r )2m}$. Recently, Li, Liu and Zhang proved the same bound for every $F$-free graph $G$, where $F$ is any color-critical graph with $\chi(F)=r+1\ge4$, with equality only for regular complete $r$-partite graphs. It is then natural to ask how many copies of $F$ are forced once $\lambda (G)$ exceeds this threshold. Fang, Lin and Zhai answered this at the threshold itself, and conjectured that for any fixed $C>0$, the condition $\lambda (G)\ge \sqrt{(1\!-\!{1}/{r})2m} +C$ forces $\Omega\!\left(m^{(f-1)/2}\right)$ copies.
In this paper, we answer this question with the best possible constant, proving that for every color-critical graph $F$ with $\chi(F)=r+1\ge4$, there exists $\delta_F>0$ such that if $m$ is sufficiently large, $0<q\le\delta_F\sqrt m$, and $G$ is an $m$-edge graph with $\lambda^2(G)\ge 2\left(1-\tfrac1r\right)m+q$, then \[
N_F(G)\ge\bigl(B_F-o(1)\bigr)\,q\,
m^{{(f-2)}/{2}},
\quad \text{where}~~
B_F:=\tfrac{\alpha_F}{4}
(\tfrac{2r}{r-1} )^{{f}/{2}}, \] and the constant $B_F$ is best possible. Our result can be viewed as an edge-spectral counterpart of Mubayi's theorem, since it converts the spectral surplus $q$ into a linear number of copies with a sharp constant, and it solves the conjecture of Fang, Lin and Zhai in a stronger form.