An edge-spectral Erdős-Stone-Simonovits theorem and its stability

Abstract: We study the extremal problem that relates the spectral radius $\lambda (G)$ of an $F$-free graph $G$ with its number of edges. Firstly, we prove that for any graph $F$ with chromatic number $\chi (F)=r+1\ge 3$, if $G$ is an $F$-free graph on $m$ edges, then $\lambda^2(G)\le {(1-\frac{1}{r} + o(1))2m}$. This provides a unified extension of both the Erd\H{o}s--Stone--Simonovits theorem and its vertex-spectral version due to Nikiforov, and confirms a conjecture proposed by Li, Liu and Feng. We also establish the corresponding edge-spectral stability, showing that if $G$ is an $F$-free graph on $m$ edges with $\lambda^2(G)=(1- \frac{1}{r} - o(1))2m$, then $G$ differs from a complete bipartite graph by $o(m)$ edges when $r=2$, and $G$ differs from an $r$-partite Tur\'{a}n graph by $o(m)$ edges when $r\ge 3$. This extends the classical Erd\H{o}s--Simonovits stability theorem. As an application of our method, we improve a result of Zhai, Lin and Shu by showing that if $\lambda (G)>\sqrt{m}$, then there exist two vertices in $G$ that have at least $\frac{1}{2}\sqrt{m} - O(1)$ common neighbors. This bound is the best possible as witnessed by a random construction.