Sharp spectral lower bounds for the booksize of a graph

RESUMO

Let $G$ be an $n$-vertex graph with $e(G)$ edges, and let $\lambda(G)$ denote the largest eigenvalue of its adjacency matrix. The booksize $\mathrm{bk} (G)$ of $G$ is defined as the largest number of triangles sharing a common edge. The main purpose of this note is to prove that if $\lambda(G)\geq\lambda(T_{n,2})$ and $G$ contains a triangle, then $\mathrm{bk} (G)\geq \lambda(G) - n/3$. To obtain this goal, we prove the sharper quantitative estimate $\mathrm{bk} (G)\geq\lambda(G)-\frac{2e(G)}{3\lambda(G)}$ whenever $\lambda(G)^2 > e(G)$. As the first consequence, we derive that an $n$-vertex graph $G$ with $e(G) > n^2/4$ satisfies that $\mathrm{bk} (G) > \frac{2e(G)}{n} - \frac{n}{3}$, which is stronger than an old conjecture of Erdős, and was proved by Edwards. The second corollary is a positive solution to an open problem by Zhai and Lin (JGT, 2023). The third application is a positive solution to an open problem of Li, Liu, and Zhang (JCTB, 2026), which was independently proved by Zhang et al.