Geodesic
Spectral integral variation of trees
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 49-58

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In this paper, we determine all trees with the property that adding a particular edge will result in exactly two Laplacian eigenvalues increasing respectively by 1 and the other Laplacian eigenvalues remaining fixed. We also investigate a situation in which the algebraic connectivity is one of the changed eigenvalues.
Keywords: tree, Laplacian eigenvalues, spectral integral variation, algebraic connectivity 
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Wang, Yi; Fan, Yi-Zheng. Spectral integral variation of trees. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 49-58. http://geodesic.mathdoc.fr/item/DMGT_2006_26_1_a4/