添加边后的覆盖时间的非局域性
Nonlocality of Cover-Time Changes Under Edge Addition
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中文总结 AI 辅助
本文研究添加边后简单随机游走的期望覆盖时间变化，通过 killed Green 矩阵更新结合目标集容斥得到精确公式，证明了覆盖时间响应的非局域性，并补充电导插值定理及路径的相关结论。
中文摘要 AI 辅助
设$G$为有限连通简单图，$uv$为一条非边，$s$为起始顶点。我们研究插入$uv$后简单随机游走的期望覆盖时间的精确变化。结合目标集容斥的 killed Green 矩阵更新，可得到仅使用原图的精确公式，该响应可正可负。我们的主要结果是一个非局域性定理：对每个半径$r\geq1$，我们构造两个标记构型，其在$s、u、v$处的环境度标记$r$邻域同构，但固定起始点的覆盖时间响应符号相反；该构造还匹配三个标记度、标记距离和有效电阻$R_{uv}$，两个图的阶数不同，等阶非局域性仍待解决。我们补充了电导插值定理及其高电导系数的占据解释：收缩$u$和$v$后，该系数为收缩顶点预覆盖期望占据的正倍数，其零情形可精确分类；对含两个悬挂插入端点的路径，求解了所有起始顶点的情况，显示其在起始集上存在尖锐符号变化。
英文摘要
Let $G$ be a finite connected simple graph, let $uv$ be a nonedge, and let $s$ be a starting vertex. We study the exact change in the expected cover time of simple random walk when $uv$ is inserted. A killed Green matrix update, combined with target-set inclusion-exclusion, gives an exact formula using only the original graph. The response can have either sign. Our main result is a nonlocality theorem. For every radius $r\geq 1$, we construct two marked configurations whose ambient-degree-labelled radius-$r$ neighbourhoods at $s,u,v$ are isomorphic but whose fixed-start cover-time responses have opposite signs. The construction also matches the three marked degrees, the marked distance, and the effective resistance $R_{uv}$. The two graphs have different orders; equal-order nonlocality remains open. We complement this result with a conductance interpolation theorem and an occupation interpretation of its high-conductance coefficient. After contracting $u$ and $v$, that coefficient is a positive multiple of the expected pre-cover occupation of the contracted vertex, and its zero case is classified exactly. A path with two pendant insertion endpoints is solved for all starting vertices and shows a sharp change of sign across the start set.
