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$成为一棵连结的树, 以美元为顶端, 让美元=D-A$, 表示拉普拉西亚矩阵以G$为单位。 第二小的egenvalue $\lambda<unk> 2}(G) > 0$, 也称为代数连接, 以及相关的egenvictor $\phi_ 2$, 引起了很大的兴趣。 我们调查了何时在最长路径的终点假定$\phi_ 2$的峰值和迷你值为$G$。 我们的结果也适用于更一般的图表, 该图表“ 在全球保持” 像树一样, 但可以展示更复杂的本地结构 。 关键的新成分是复制 Egenvictor $\phi_k$的公式 。</s>