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. Finally, a recursive formula for computing the Hosoya-Wiener polynomial of a weighted tree is given.

In this paper, the Wiener Index ( ) ( ) { } ( ) ∑ ∈ = G V u v u v d G W , , and Hosoya polynomial ( ) ( ) { } ( ) ∑ ∈ = G V u v u v d x x G H , , , of a class of Jahangir graphs m J , 3 with exactly 1 3 + m vertices and m 4 edges are computed.

The Merrifield-Simmons index is related to several physicochemical characteristics and is thus of use in combinatorial chemistry, e.g. in drug design and molecular recognitions. In this paper, we show how one can algorithmically construct databases of acyclic molecular graphs with large Merrifield-Simmons index. Our algorithm can deal with a large number of atoms (several hundreds) in short tim...