A WIENER-TYPE GRAPH INVARIANT FOR SOME BIPARTITE GRAPHS

In this paper, a Wiener-type graph invariant W is considered, defined as the sum of the product n(u)(e)n(v)(e) over all edges e = (u, v) of a connected graph G, where n(u)(e) is the number of vertices of G, ly ing closer to u than to v. A class C(h, k) of bipartite graphs with cy clomatic number h is designed, such that for G(1), G(2) is an element of C(h, k), W(G(1)) = W*(G(2)) (mod 2k(2)). This fully parallels a pr eviously known result for the Wiener number.