On the distribution of Laplacian eigenvalues of trees. (English) Zbl 1279.05045
Summary: For a tree \(T\) with \(n\) vertices, we apply an algorithm due to D. P. Jacobs and V. Trevisan [Linear Algebra Appl. 434, No. 1, 81–88 (2011; Zbl 1231.05167)] to study how the number of small Laplacian eigenvalues behaves when the tree is transformed by a transformation defined by B. Mohar [ibid. 422, No. 2–3, 736–741 (2007; Zbl 1120.05055)]. This allows us to obtain a new bound for the number of eigenvalues that are smaller than 2. We also report our progress towards a conjecture on the number of eigenvalues that are smaller than the average degree.
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