The minimal Laplacian spectral radius of trees with a given diameter

For a graph G, its Laplacian matrix is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. Let Tn,d be the set of trees on n vertices with diameter d. In this paper, for d∈{1,2,3,4,n−3,n−2,n−1}, trees with minimal Laplacian spectral radii in the set Tn,d are characterized.