WIENER NUMBER OF VERTEX-WEIGHTED GRAPHS AND A CHEMICAL APPLICATION

The Wiener number W(G) of a graph G is the sum of distances between al l pairs of vertices of G. If (G, w) is a vertex-weighted graph, then t he Wiener number W(G, w) of (G, w) is the sum, over all pairs of verti ces, of products of weights of the vertices and their distance. For G being a partial binary Hamming graph, a formula is given for computing W(G, w) in terms of a binary Hamming labelling of G. This result is a pplied to prove that W(PH) = W((HS) over tilde) + 36W(ID), where PH is a phenylene, (HS) over tilde a pertinently vertex-weighted hexagonal squeeze of PH, and ID the inner dual of the hexagonal squeeze.