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Let $L\in \mathbb{R}^{n \times n}$ be the Laplacian matrix of a simple undirected graph and $D_i$ be the same size matrix with $i$th diagonal element $1$. Denote the smallest eigenvalue of $L+D_i$ as $\lambda_1(i)$ and related normalized eigenvector $v_1(i)\in \mathbb{R}^n$. Then, how to show that for $\forall j\ne i$ such that $|v_{1j}(i)|\ge|v_{1i}(i)|$, i.e., the $i$th entry in $v_1(i)$ has the smallest magnitude.

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