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：让$G$成为一棵连接的树, 以美元为顶端, 让$L = D- A$ 表示拉普拉西亚矩阵以$G$为单位。 第二小的乙基值$\lambda ⁇ 2}(G) > 0$, 也称为代数连接, 以及相关的乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙型乙