Wiener number of vertex-weighted graphs and a chemical application

Abstract

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