On the differences between Szeged and Wiener indices of graphs

Let G be a connected graph and eta(G) = Sz(G) - W(G), where W(G) and Sz(G) are the Wiener and Szeged indices of G, respectively. A well-known result of Klavzar, Rajapakse, and Gutman states that eta(G) >= 0, and by a result of Dobrynin and Gutman eta(G) = 0 if and only if each block of G is complete. In this paper, a path-edge matrix for the graph G is presented by which it is possible to classify the graphs in which eta(G) = 2. It is also proved that there is no graph G with the property that eta(G) = 1 or eta(G) = 3. Finally, it is proved that, for a given positive integer k, k not equal 1. 3, there exists a graph G with eta(G) = k. (C) 2011 Elsevier B.V. All rights reserved.