Note on the kth Laplacian eigenvalues of trees with perfect matchings

Denote by μk(T)μk(T) the kth Laplacian eigenvalue of tree T  . Let TnTn be the set of all trees of order n with perfect matchings. In this paper, we characterize the tree T   in T4tT4t with μ2(T)=t+2+t2+42, and the tree in T6tT6t with μ3(T)=t+2+t2+42. These results affirm Problem of J.X. Li, W.C. Shiu, A. Chang (2010) [5] for the cases k=2k=2 and k=3k=3, respectively.