Laplacian energy of diameter 3 trees

Let T n be the set of all trees of diameter 3 and n vertices. We show that the Laplacian energy of any tree in T n is strictly between the Laplacian energy of the path P n and the star S n , partially proving the conjecture that this holds for any tree. We also give a total order by the Laplacian energy in T n . Moreover, we show that this order depends only on the algebraic connectivity of the tree: the Laplacian energy increases as the algebraic connectivity decreases in T n .