Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $\lambda_{2}(G) > 0$, also known as the algebraic connectivity, as well as the associated eigenvector $\phi_2$ have been of substantial interest. We investigate the question of when the maxima and minima of $\phi_2$ are assumed at the endpoints of the longest path in $G$. Our results also apply to more general graphs that `behave globally' like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for the eigenvector $\phi_k$.

翻译：设 $G$ 为包含 $n$ 个顶点的连通树，$L = D-A$ 表示 $G$ 的拉普拉斯矩阵。第二小特征值 $\lambda_{2}(G) > 0$（即代数连通度）及其对应的特征向量 $\phi_2$ 一直备受关注。我们研究了 $\phi_2$ 的最大值和最小值是否在 $G$ 中最长路径的端点处取到的问题。本文结果也适用于那些“全局行为”类似树但可能具有更复杂局部结构的图。关键的创新点在于引入了特征向量 $\phi_k$ 的再生公式。