On the Laplacian coefficients of trees with a perfect matching

Let T be a tree with n vertices and let be the characteristic polynomial of Laplacian matrix of T. It is well known that cn-2(T) is equal to the Wiener index of T, while cn-3(T) is equal to the modified hyper-Wiener index of T. For two n-vertex trees T1 and T2, write T1⪯T2 if ck(T1)⩽ck(T2) for all 0⩽k⩽n. Let Γ(n) be the set of all trees with 2n vertices and perfect matchings except P2n and An-1,1, where Ps is a path with s vertices and As,t is a tree obtained from a star St+1 with t+1 vertices by attaching s pendent paths of length 2 to the center of St+1 . In this paper, we first give a graphic transformation that increases all Laplacian coefficients of an arbitrary tree, we then determine the minimum element and the second minimum element in Γ(n) under the partial order ⪯, and we finally identify the maximum element and second maximum element in Γ(n) under the partial order ⪯. Furthermore, we characterize some trees with extremal Wiener indices and Laplacian-like energies.