Modified hyper-Wiener index of trees

For a tree T of order n with Laplacian eigenvalues μ1≥μ2≥⋯≥μn−1>μn=0, the Wiener index and the modified hyper-Wiener index are defined as [Formula presented]respectively. The modified hyper-Wiener index of T is also expressed in terms of WT and Laplacian eigenvalues as [Formula presented]In the present paper, we establish some lower and upper bounds for WWW(T).