The Laplacian spectral radius of trees and maximum vertex degree

Let Δ ( T ) and μ ( T ) denote the maximum degree and the Laplacian spectral radius of a tree T , respectively. In this paper we prove that for two trees T 1 and T 2 on n ( n ≥ 21 ) vertices, if Δ ( T 1 ) > Δ ( T 2 ) and Δ ( T 1 ) ≥ ⌈ 11 n 30 ⌉ + 1 , then μ ( T 1 ) > μ ( T 2 ) , and the bound “ Δ ( T 1 ) ≥ ⌈ 11 n 30 ⌉ + 1 ” is the best possible. We also prove that for two trees T 1 and T 2 on 2 k ( k ≥ 4 ) vertices with perfect matchings, if Δ ( T 1 ) > Δ ( T 2 ) and Δ ( T 1 ) ≥ ⌈ k 2 ⌉ + 2 , then μ ( T 1 ) > μ ( T 2 ) .