Fang–Lin–Zhai spectral supersaturation conjecture for color-critical graphs

Let $F$ be a color-critical graph with order $f=|F|$ and chromatic number $\chi(F)=r+1\geqslant4$. Write $N_F(G)$ for the number of copies of $F$ in an $m$-edge graph $G$, and let $\lambda(G)$ denote its adjacency spectral radius. Fang–Lin–Zhai conjecture. For any fixed positive constant $C$ and sufficiently large $m$,

$$\lambda(G)\geqslant\sqrt{\Bigl(1-\frac{1}{r}\Bigr)2m}+C\quad\Rightarrow\quad N_F(G)=\Omega\!\left(m^{\frac{f-1}{2}}\right).$$

This conjecture asks for a supersaturation result when the spectral radius exceeds the color-critical threshold by a fixed additive amount. Fang, Lin and Zhai proved the sharp-order threshold result at $\lambda^2(G)\geqslant(1-1/r)2m$, apart from regular complete $r$-partite graphs; the stronger additive-gap bound remains open.