Wiener number of vertex-weighted graphs and a chemical application (Q1377617)

The Wiener number of a weighted, connected graph \((G,w)\) is defined as \[ W(G,w)= \sum_{\{u,v\} \subseteq V(G)} w(u)w(v) d_G(u,v) \] where \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). A graph \(G\) is called a partial binary Hamming graph if each vertex \(v\) can be labelled by a word of fixed length, \(\ell(v)\), defined on the alphabet \(\{0,1\}\) such that for all \(u,v\in V(G)\) the distance between \(u\) and \(v\) is the Hamming distance between \(\ell(u)\) and \(\ell(v)\). A formula is developed for computing \(W(G,w)\) for such graphs \(G\). This formula is then used to express the Wiener number of the chemical graphs associated with phenylenes in terms of certain simpler related graphs.