Weighted Wiener number of trees
The Wiener index is the first non-trivial topological index in chemistry that was introduced by H. Wiener in 1974 to study boiling points of paraffins. Formulas for the Wiener number and the Hosoya-Wiener polynomial of edge and vertex weighted graphs are given in terms of edge and path contributions. For a rooted tree, the Hosoya-Wiener polynomial is expressed as a sum of vertex contributions. Efficient methods for computing the Wiener number and the Hosoya-Wiener polymomial are outlined.