Li–Peng square-root booksize conjecture for Nosal graphs

Let $G$ be an $m$-edge graph with spectral radius $\rho(G)>\sqrt{m}$. Such a graph is called Nosal, and let $bk(G)$ denote the maximum number of triangles sharing a common edge in $G$.

Li–Peng conjecture. For every $m$-edge Nosal graph $G$,

$$bk(G)=\Omega(\sqrt{m}).$$

Nikiforov had previously shown the weaker bound $bk(G)>\frac{1}{12}\sqrt[4]{m}$. The conjecture was confirmed by Li, Liu and Zhang, who proved $bk(G)>\frac{1}{24}\sqrt{m}$ and showed that the order $\sqrt{m}$ is best possible.