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(HS) + 36W(ID), where PH is a phenylene, HS a pertinently vertex-weighted hexagonal squeeze of PH, and ID the inner dual of the hexagonal squeeze.

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