The Laplacian spectral radius of trees and maximum vertex degree

Let Δ(T)Δ(T) and μ(T)μ(T) denote the maximum degree and the Laplacian spectral radius of a tree TT, respectively. In this paper we prove that for two trees T1T1 and T2T2 on n(n≥21) vertices, if Δ(T1)>Δ(T2)Δ(T1)>Δ(T2) and Δ(T1)≥⌈11n30⌉+1, then μ(T1)>μ(T2)μ(T1)>μ(T2), and the bound “Δ(T1)≥⌈11n30⌉+1” is the best possible. We also prove that for two trees T1T1 and T2T2 on 2k(k≥4) vertices with perfect matchings, if Δ(T1)>Δ(T2)Δ(T1)>Δ(T2) and Δ(T1)≥⌈k2⌉+2, then μ(T1)>μ(T2)μ(T1)>μ(T2).