On the Laplacian spectral radii of trees

Let Δ(T)Δ(T) and μ(T)μ(T) denote the maximum degree and the Laplacian spectral radius of a tree TT, respectively. Let TnTn be the set of trees on nn vertices, and Tnc={T∈Tn∣Δ(T)=c}. In this paper, we determine the two trees which take the first two largest values of μ(T)μ(T) of the trees TT in Tnc when c≥⌈n2⌉. And among the trees in Tnc, the tree which alone minimizes the Laplacian spectral radius is characterized. We also prove that for two trees T1T1 and T2T2 in Tn(n≥6), if Δ(T1)>Δ(T2)Δ(T1)>Δ(T2) and Δ(T1)≥⌈n2⌉+1, then μ(T1)>μ(T2)μ(T1)>μ(T2). As an application of these results, we give a general approach about extending the known ordering of trees in TnTn by their Laplacian spectral radii.