On the Laplacian spectral radii of trees

Let Delta(T) and mu(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 is an element of T-n | Delta(T) = c}. In this paper, we determine the two trees which take the first two largest values of mu(T) of the trees T in T-n(c) when c > inverted right perpendicular n/2 inverted left perpendicular. 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 Delta(T-1) > Delta(T-2) and Delta(T-1) >= inverted right perpendicular n/2 inverted left perpendicular + 1, then mu(T-1) > mu(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. (C) 2009 Published by Elsevier B.V.