AI 中文总结
本文针对连通图的归一化距离拉普拉斯谱间隙问题，证明了比2025年猜想更强的下界，明确了等号成立的图结构条件，完善了该领域的理论结论。
AI 中文摘要
连通图的归一化距离拉普拉斯矩阵$\boldsymbol{\textit{D}}^{\boldsymbol{\textit{L}}}$的最小正特征值$\boldsymbol{\textit{\text{∂}}}_2$被称为其谱间隙，与$\boldsymbol{\textit{D}}^{\boldsymbol{\textit{L}}}$的Cheeger常数密切相关。Byrne、Johnston、Schildkraut和Tait于2025年提出猜想：所有连通图均满足$\boldsymbol{\textit{\text{∂}}}_2 \boldsymbol{\textit{≥}} \frac{2}{3}$。本文证明了更强的结果：对于任意阶数至少为2的连通图$\boldsymbol{\textit{G}}$，有$\boldsymbol{\textit{\text{∂}}}_2 \boldsymbol{\textit{≥}} \frac{2}{3} + \frac{4}{3\boldsymbol{\textit{t}}_{\text{max}}}$，其中$\boldsymbol{\textit{t}}_{\text{max}}$表示$\boldsymbol{\textit{G}}$中的最大传输；等号成立当且仅当$\boldsymbol{\textit{G}} \boldsymbol{\textit{≅}} \boldsymbol{\textit{K}}_{\boldsymbol{\textit{m,m}}}$（$\boldsymbol{\textit{m}} \boldsymbol{\textit{≥}} 1$）。
英文摘要
The smallest positive eigenvalue $\partial_2$ of the normalized distance Laplacian matrix $\mathcal{D}^{\mathcal{L}}$ of a connected graph is called its \emph{spectral gap} and is intimately related to the Cheeger constant of $\mathcal{D}^{\mathcal{L}}$. Byrne, Johnston, Schildkraut and Tait (2025) conjectured that \[ \partial_2 \ge \frac{2}{3}\] for all connected graphs. We prove the following stronger result: for any connected graph $G$ of order at least 2, \[\partial_2 \ge \frac{2}{3} + \frac{4}{3\,t_{\max}},\] where $t_{\max}$ denotes the maximum transmission in $G$. Moreover, equality holds if and only if $G\cong K_{m,m}$ for some $m\ge 1$.
