A Wiener-type graph invariant for some bipartite graphs (Q1904516)

The Wiener number (index) of a connected graph \(G\) is the sum \(W(G)\) of distances between all pairs of vertices of \(G\). In this paper, a similar invariant is considered: the sum \(W^*(G)\) of the products \(n_u(e)n_v(e)\) over all edges \(e= (u, v)\), where \(n_u(e)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\). A class of bipartite graphs is shown to have the same property for \(W\) and \(W^*\).