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 all pairs of vertices of G. If (G,w) is a vertex-weighted graph, then the Wiener number W (G,w) of (G,w) is the sum, over all pairs of vertices, 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 labeling of G. This result is applied to prove that W (PH) = W (H̃S) + 36W (ID) , where PH is a phenylene, H̃S a pertinently vertex-weighted hexagonal squeeze of PH , and ID the inner dual of the hexagonal squeeze.