图的booksize的尖锐谱下界
The spectral Erdős book theorem: sharp bounds and stability
- Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
- School of Mathematical Sciences, Anhui University（安徽大学数学科学学院）
- College of Cryptology and Cyber Science & College of Computer Science, Nankai University（南开大学密码学与网络空间安全学院及计算机学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了图的booksize的尖锐谱下界，即当最大特征值超过某阈值且图含三角形时，booksize至少为特征值减n/3，并由此解决了多个开放问题。
AI中文摘要：
设$G$是一个具有$n$个顶点和$e(G)$条边的图，$\lambda(G)$表示其邻接矩阵的最大特征值。$G$的booksize $\mathrm{bk}(G)$定义为共享一条公共边的三角形的最大数量。本文的主要目的是证明：如果$\lambda(G)\geq\lambda(T_{n,2})$且$G$包含一个三角形，则$\mathrm{bk}(G)\geq\lambda(G)-n/3$。为实现这一目标，我们证明了更尖锐的定量估计：当$\lambda(G)^2>e(G)$时，$\mathrm{bk}(G)\geq\lambda(G)-\frac{2e(G)}{3\lambda(G)}$。作为第一个推论，我们得出：一个具有$n$个顶点且$e(G)>n^2/4$的图$G$满足$\mathrm{bk}(G)>\frac{2e(G)}{n}-\frac{n}{3}$，这强于Erdős的一个旧猜想，且该猜想已被Edwards证明。第二个推论是对Zhai和Lin（JGT, 2023）提出的一个开放问题的肯定解答。第三个应用是对Li、Liu和Zhang（JCTB, 2026）提出的一个开放问题的肯定解答，该问题已被Zhang等人独立证明。
英文摘要：
The booksize $\mathrm{bk}(G)$ of a graph $G$ is the largest number of triangles sharing a common edge. A classical theorem of Edwards, conjectured by Bollobás and Erdős, states that every $n$-vertex graph $G$ with $e(G)>e(T_{n,2})$ has booksize greater than $n/6$. Zhai and Lin [J. Graph Theory 102 (2023) 502--520] asked whether the same conclusion holds under the spectral condition $λ(G)>λ(T_{n,2})$, where $λ(G)$ is the spectral radius of the adjacency matrix. We answer this question in a strong form: every $n$-vertex graph $G\neq T_{n,2}$ with $λ(G)\geλ(T_{n,2})$ satisfies \[ \mathrm{bk}(G)\ge\max\Big\{\frac13λ(G),\,λ(G)-\frac n3,\,2λ(G)-n\Big\}. \] Consequently, the condition $λ(G)>λ(T_{n,2})$ forces $\mathrm{bk}(G)\ge\lfloor n/6\rfloor+1$. The middle term is a spectral improvement of Edwards' bound $\mathrm{bk}(G)\ge\frac{2m}{n}-\frac n3$, and all three bounds are best possible. These results come from the edge-spectral setting: every graph $G$ with $m$ edges and $λ(G)\ge\sqrt m$ that is not a complete bipartite graph satisfies \[ \mathrm{bk}(G)\ge\max\Big\{λ(G)-\frac{2m}{3λ(G)},\,2λ(G)-\frac{2m}{λ(G)}\Big\}, \] which strengthens the bound $\mathrm{bk}(G)\ge\frac13λ(G)$ of Zhao, You, Zeng and Zhang. As an application of our method, we prove a triangle counting bound $t(G)\ge\frac13(λ(G)+1)(λ(G)^2-m)$, which improves the result of Bollobás and Nikiforov [J. Combin. Theory Ser B. (2007)]. Finally, we prove stability results at both thresholds: if $λ(G)\ge(\frac12-o(1))n$, then either $\mathrm{bk}(G)\ge(\frac16-o(1))n$ or $G$ can be made into $T_{n,2}$ by adding and deleting $o(n^2)$ edges; if $λ(G)\ge(1-o(1))\sqrt m$, then either $\mathrm{bk}(G)\ge(\frac13-o(1))\sqrt m$ or $G$ differs from a complete bipartite graph in $o(m)$ edges.
