刻画拉普拉斯特征值的布劳威尔不等式中的等式情形
Characterizing the equality case in Brouwer's inequality for Laplacian eigenvalues
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中文总结 AI 辅助
研究刻画拉普拉斯特征值的布劳威尔不等式中等式情形，核心方法是依赖科塔里和图多塞的投影方法，主要贡献是得出等式成立当且仅当图为团数为\(k + 1\)的阈值图，为完整布劳威尔猜想提供完整解决方案。
中文摘要 AI 辅助
布劳威尔猜想，对于每个具有\(n\)个顶点的图以及每个\(k\in \{1,2,\dots,n\}\)，其\(k\)个最大拉普拉斯特征值之和小于或等于边数加上\(\binom{k + 1}{2}\)，最近已被科塔里和图多塞（2026年）证实。在本笔记中，我们刻画了该不等式中的等式情形。主要结果是，对于每个\(n\)顶点图\(G=(V,E)\)和每个\(k\in \{1,2,\dots,n - 1\}\)，等式\(\sum_{i = 1}^k\mu_i(G)=|E(G)|+\binom{k + 1}{2}\)成立当且仅当\(G\)是团数为\(k + 1\)的阈值图，其中\(\mu_1(G)\geq \mu_2(G)\geq \cdots\geq \mu_{n}(G)\)是\(G\)的拉普拉斯特征值。这与已证实的布劳威尔猜想一起，将为李和郭（2022年）提出的完整布劳威尔猜想提供完整解决方案。我们的证明依赖于科塔里和图多塞的投影方法，并直接表明等式情形仅在阈值图中出现。
英文摘要
Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for every $k\in \{1,2,\dots,n\}$, which has been confirmed by Kothari and Tudose (2026) recently. In this note, we characterize the equality case in this inequality. Our main result is that for every $n$-vertex graph $G=(V,E)$ and for every $k\in \{1,2,\dots,n-1\}$, the equality $\sum_{i=1}^kμ_i(G)=|E(G)|+\binom{k+1}{2}$ holds if and only if $G$ is a threshold graph with clique number $k+1$, where $μ_1(G)\geq μ_2(G)\geq \cdots\geq μ_{n}(G)$ are the Laplacian eigenvalues of $G$. This, together with the confirmed Brouwer's conjecture, would yield a complete solution to the full Brouwer's conjecture posed by Li and Guo (2022). Our proof relies on the projection method of Kothari and Tudose and shows directly that the equality case can occur only for threshold graphs.
