### Abstract

Let ∆(

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

Journal | Graph Theory Notes of New York |

Volume | 58 |

Publication status | Published - 2010 |