The second smallest eigenvalue of the Laplacian matrix of a (connected) graph is known as the algebraic connectivity of a graph and the corresponding eigenvectors are known as Fiedler vectors. I got the idea of monotonicity properties of Fiedler vectors of a tree from the book - 'Graphs and matrices' by R. B. Bapat. The results are obtained only for simple undireced trees without having edge-weights. I expect the results are still valid for the weighted graphs. So my question is - Is there any work related to the Fiedler vectors of the weighted graph case? If so please suggest some reference.
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I got it.
'A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory'-by Miroslav Fiedler, Czechoslovak Mathematical Journal, Vol. 25 (1975), No. 4, 619--633
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