• Corpus ID: 209405299

Extreme Values of the Fiedler Vector on Trees

@article{Lederman2019ExtremeVO,
  title={Extreme Values of the Fiedler Vector on Trees},
  author={Roy R. Lederman and Stefan Steinerberger},
  journal={ArXiv},
  year={2019},
  volume={abs/1912.08327}
}
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… 
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