AI 中文总结
研究关于布劳威尔拉普拉斯猜想扩展的两个猜想，利用布劳威尔拉普拉斯不等式证明了Lew提出的两个关于最大拉普拉斯特征值之和上界的猜想。
AI 中文摘要
设\(G=(V,E)\)是一个阶为\(n\)的简单图，\(\lambda_1(G)\geq\cdots\geq\lambda_n(G)\)是其拉普拉斯矩阵的特征值。布劳威尔猜想对于每个\(1\leq k\leq n\)，\(\sum_{i = 1}^k\lambda_i(G)\leq|E|+\binom{k + 1}{2}\)。Lew建立了布劳威尔拉普拉斯特征值不等式的较弱形式，Kothari和Tudose最近证明了完整的布劳威尔猜想。Lew还提出了两个关于最大拉普拉斯特征值之和上界的猜想，本文利用布劳威尔拉普拉斯不等式证明了这两个猜想。
英文摘要
Let $G=(V,E)$ be a simple graph of order $n$ and let $λ_1(G)\ge \cdots \ge λ_n(G)$ be the eigenvalues of its Laplacian matrix. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^kλ_i(G)\le |E|+\binom{k+1}{2}$, which was recently confirmed by Kothari and Tudose. Before Brouwer's conjecture was proved, Lew (JCT-B, 2026) established a weaker form of Brouwer's Laplacian eigenvalue inequality and proposed two conjectures for upper bounds on the sum of the $k$ largest Laplacian eigenvalues, one in terms of the matching number and the other in terms of the vertex-cover number. Using Brouwer's Laplacian inequality, we prove both conjectures.
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