Extreme values of the Fiedler vector on trees

Abstract

Let G be a tree on n vertices and let $L=D-A$ denote the Laplacian matrix on G. The second-smallest eigenvalue ${\lambda }_2\left(G\right)>0$, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector 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 eigenvectors of graphs.