On the Laplacian spectral radii of trees

Let Δ ( T ) and μ ( T ) denote the maximum degree and the Laplacian spectral radius of a tree T , respectively. Let T n be the set of trees on n vertices, and T n c = { T ∈ T n ∣ Δ ( T ) = c } . In this paper, we determine the two trees which take the first two largest values of μ ( T ) of the trees T in T n c when c ≥ ⌈ n 2 ⌉ . And among the trees in T n c , the tree which alone minimizes the Laplacian spectral radius is characterized. We also prove that for two trees T 1 and T 2 in T n ( n ≥ 6 ) , if Δ ( T 1 ) > Δ ( T 2 ) and Δ ( T 1 ) ≥ ⌈ n 2 ⌉ + 1 , then μ ( T 1 ) > μ ( T 2 ) . As an application of these results, we give a general approach about extending the known ordering of trees in T n by their Laplacian spectral radii.