A Wiener-type graph invariant for some bipartite graphs

Abstract

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, lying closer to u than to v. A class C( h, k) of bipartite graphs with cyclomatic number h is designed, such that for G 1, G 2 ∈ C( h, k), W ∗(G 1) ≡ W ∗(G 2) ( mod 2k 2) . This fully parallels a previously known result for the Wiener number.