Lederman, Roy R; Steinerberger, S Extreme Values of the Fiedler Vector on Trees Technical Report 2019, (arXiv: 1912.08327). Abstract | Links | BibTeX | Tags: Computer Science - Discrete Mathematics, Graph Theory, Mathematics - Combinatorics, Mathematics - Spectral Theory @techreport{lederman_extreme_2019, title = {Extreme Values of the Fiedler Vector on Trees}, author = {Roy R Lederman and S Steinerberger}, url = {http://arxiv.org/abs/1912.08327}, year = {2019}, date = {2019-12-01}, urldate = {2020-08-13}, abstract = {Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $textbackslashlambda_2(G) textgreater 0$, also known as the algebraic connectivity, as well as the associated eigenvector $textbackslashphi_2$ have been of substantial interest. We investigate the question of when the maxima and minima of $textbackslashphi_2$ 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 $textbackslashphi_k$.}, note = {arXiv: 1912.08327}, keywords = {Computer Science - Discrete Mathematics, Graph Theory, Mathematics - Combinatorics, Mathematics - Spectral Theory}, pubstate = {published}, tppubtype = {techreport} } Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $textbackslashlambda_2(G) textgreater 0$, also known as the algebraic connectivity, as well as the associated eigenvector $textbackslashphi_2$ have been of substantial interest. We investigate the question of when the maxima and minima of $textbackslashphi_2$ 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 $textbackslashphi_k$. |