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

Abstract

We study the supersaturation problem in its edge-spectral form. Let $λ(G)$ be the adjacency spectral radius of $G$. Nikiforov proved that every $K_{r+1}$-free graph $G$ with $m$ edges satisfies $λ(G)\le \sqrt{(1\!-\!1/r )2m}$. Recently, Li, Liu and Zhang proved the same bound for every $F$-free graph $G$, where $F$ is any color-critical graph with $χ(F)=r+1\ge4$, with equality only for regular complete $r$-partite graphs. It is then natural to ask how many copies of $F$ are forced once $λ(G)$ exceeds this threshold. Fang, Lin and Zhai answered this at the threshold itself, and conjectured that for any fixed $C>0$, the condition $λ(G)\ge \sqrt{(1\!-\!{1}/{r})2m} +C$ forces $Ω\!\left(m^{(f-1)/2}\right)$ copies. In this paper, we answer this question with the best possible constant. Building on the proof framework of Fang, Lin and Zhai, we prove that for every color-critical graph $F$ with $χ(F)=r+1\ge4$, there exists $δ_F>0$ such that if $m$ is sufficiently large, $0<q\leδ_F\sqrt m$, and $G$ is an $m$-edge graph with $λ^2(G)\ge 2\left(1-\tfrac1r\right)m+q$, then \[ N_F(G)\ge\bigl(B_F-o(1)\bigr)\,q\, m^{{(f-2)}/{2}}, \quad \text{where}~~ B_F:=\tfrac{α_F}{4} (\tfrac{2r}{r-1} )^{{f}/{2}}, \] 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 of $F$, and it solves the conjecture of Fang, Lin and Zhai in a stronger form.