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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.