AI 中文总结
研究布劳威尔拉普拉斯猜想，通过证明特定条件下等式成立的充要条件是图为团数为\(k + 1\)的阈值图，证实了相关猜想。
AI 中文摘要
布劳威尔拉普拉斯猜想断言，对于任何具有\(n\)个顶点和\(m\)条边的图\(G\)，\(k\)个最大拉普拉斯特征值之和满足\(s_k(G) \le m + \binom{k + 1}{2}\)，\(k = 1, \ldots, n\)。本文证明了对于某些\(1\le k\le n - 1\)等式成立当且仅当\(G\)是团数为\(k + 1\)的阈值图，证实了李和郭提出的完全布劳威尔猜想。
英文摘要
The Laplacian conjecture of Brouwer asserts that for any graph \(G\) of order n with \(m\) edges, the sum of the \(k\) largest Laplacian eigenvalues satisfies \(s_k(G) \le m + \binom{k+1}{2}\) for $k=1, \ldots, n$. Later, Li and Guo in 2022 further proposed the full Brouwer's Laplacian spectrum conjecture. Recently, Kothari and Tudose in 2026 proved the Brouwer's conjecture. Motivated by their perfect proof and methods, we proved that for a simple graph of order $n$ with $m$ edges and $1\le k\le n-1$, \(s_k(G) = m + \binom{k+1}{2}\) if and only if $G$ is a threshold graph with clique number \(k+1\), which confirms the full Brouwer conjecture proposed by Li and Guo.
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