AI 中文总结
研究格罗内 - 梅里斯不等式等式成立的条件，借助科塔里和图多塞的分裂图迹不等式及蔡等人对布劳威尔等式情形的刻画，证明等式成立当且仅当图\(G\)属于两个特定族之一，给出界紧的所有对\((G,k)\)的完整描述。
AI 中文摘要
格罗内 - 梅里斯不等式由格罗内和梅里斯于1994年提出，白于2011年首次证明，即对于阶数为\(n\)的图\(G\)及\(1\leq k\leq n\)，有\(\sum_{i = 1}^k\lambda_i(G)\leq\sum_{i = 1}^k d_i^*(G)\)。本文确定了等式成立的条件。利用科塔里和图多塞在证明布劳威尔拉普拉斯猜想时所发展的分裂图迹不等式，并结合蔡、陈、杨和张对布劳威尔等式情形的刻画，证明了格罗内 - 梅里斯不等式中等式成立当且仅当图\(G\)属于两个明确描述的族之一。这两个族是通过在一个终端块上进行手术操作从阈值图得到的：第一个族是从初始支配块中移除边；第二个族是在初始孤立块内添加边。我们的分析给出了格罗内 - 梅里斯界紧的所有对\((G,k)\)的完整组合描述。
英文摘要
The Grone--Merris inequality, conjectured by Grone and Merris~(1994) and first proved by Bai~(2011), states that for every graph $G$ of order $n$ and every $1\le k\le n$, $\sum_{i=1}^kλ_i(G)\le\sum_{i=1}^k d_i^*(G)$, where $λ_1\ge\cdots\geλ_n$ are the Laplacian eigenvalues and $d_1^*\ge\cdots\ge d_n^*$ is the conjugate degree sequence. In this paper we determine exactly when equality holds. Using the split-graph trace inequality developed by Kothari and Tudose~(2026) in their proof of Brouwer's Laplacian conjecture---which relies on Bai's theorem and also establishes the equivalence between the two conjectures---together with the recent characterization of the Brouwer equality cases by Cai, Chen, Yang and Zhang~(2027), we prove that equality holds in the Grone--Merris inequality if and only if the graph $G$ belongs to one of two explicitly described families. Both families are obtained from a threshold graph by a surgical operation at one terminal block: in the first family, edges are removed from the initial dominating block; in the second, edges are added inside the initial isolated block. Our analysis yields a complete combinatorial description of all pairs $(G,k)$ for which the Grone--Merris bound is tight.
Comments16 pages
