发表机构
Shanghai Jiao Tong University（上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了在阶数至少为200的树中，平衡双彗星达到最小邻接谱隙，并给出了马尔可夫链逆谱隙与有效电阻-方差量的精确关系。
AI 中文摘要
对于一棵树 $T$，设 $g(T)=\lambda_1(T)-\lambda_2(T)$ 为其两个最大邻接特征值之差。平衡双彗星是通过在一条路径的两个端点分别连接相同数量的叶子而得到的树。Jovović、Koledin 和 Stanić 猜想，在任意固定阶数的树中，这种树达到最小的邻接谱隙。我们证明，对于阶数 $n\ge200$ 的每一棵最小化树都是平衡双彗星。我们还证明，对于任何有限不可约可逆连续时间马尔可夫链，其逆谱隙与两个状态之间的有效电阻乘以它们击中概率的平稳方差之差，至多等于在该状态处杀死的逆狄利克雷隙。对于 Perron 链，我们给出了该电阻-方差量的精确二顶点 Schur 补公式。
英文摘要
For a tree $T$, let $g(T)=λ_1(T)-λ_2(T)$ be the difference between its two largest adjacency eigenvalues. A balanced double comet is obtained by attaching equally many leaves to the two endpoints of a path. Jovović, Koledin and Stanić conjectured that such a tree attains the minimum adjacency spectral gap among trees of any fixed order. We prove that every minimizing tree of order $n\ge200$ is a balanced double comet. We also show that, for any finite irreducible reversible continuous-time Markov chain, the inverse spectral gap differs from the effective resistance between two states times the stationary variance of their hitting probability by at most the inverse Dirichlet gap for killing at those states. For Perron chains, we give an exact two-vertex Schur-complement formula for this resistance--variance quantity.
Comments22 pages
