A Wiener-type graph invariant for some bipartite graphs

AbstractIn this paper, a Wiener-type graph invariant W∗ is considered, defined as the sum of the product nu(e)nv(e) over all edges e = (u, v) of a connected graph G, where nu(e) is the number of vertices of G, lying closer to u than to v. A class C(h, k) of bipartite graphs with cyclomatic number h is designed, such that for G1, G2 ∈ C(h, k), W∗(G1) ≡ W∗(G2) (mod 2k2). This fully parallels a previously known result for the Wiener number