Ordering trees by their largest laplacian eigenvalues

Jianxi LI, Wai Chee SHIU, Wai Hong CHAN

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Let ∆(T) and μ₁(T) , respectively, denote the maximum degree and largest Laplacian eigenvalue of a tree T. Let Ƭn be the set of trees of order n, and let Ƭn(∆) = { T ϵ Ƭn : ∆(T) = ∆}. In this paper, among all trees in Ƭn(∆), we characterize the tree that minimizes the largest Laplacian eigenvalue, as well the tree that maximizes the largest Laplacian eigenvalue when n ─ 1 ≥ ∆ ≥[n/2]. Furthermore, we prove that, for two trees T₁ and T₂ in Ƭn, if ∆( T₁) ≥[2n/3] ─ 1 and ∆( T₁) ˃ ∆( T₂), then μ₁(T₁) ˃ μ₁(T₂). Using this result, we extend the order of trees in Ƭn by their largest Laplacian eigenvalues to the 13th tree when n ≥ 15. This extends the results of Guo and Yu et al. Copyright © 2010 The Mathematical Association of America.
Original languageEnglish
Pages (from-to)9-17
JournalGraph Theory Notes of New York
Publication statusPublished - 2010


Li, J., Shiu, W. C., & Chan, W. H. (2010). Ordering trees by their largest laplacian eigenvalues. Graph Theory Notes of New York, 58, 9-17.

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