Extreme Values of the Fiedler Vector on Trees
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Let G be a connected tree on n vertices and let L = D-A denote the Laplacian matrix on G. The second-smallest eigenvalue λ_2(G) > 0, also known as the algebraic connectivity, as well as the associated eigenvector ϕ_2 have been of substantial interest. We investigate the question of when the maxima and minima of ϕ_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 ϕ_k.
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