On the differences between Szeged and Wiener indices of graphs

Let G be a connected graph and η ( G ) = S z ( G ) − W ( G ) , where W ( G ) and S z ( G ) are the Wiener and Szeged indices of G , respectively. A well-known result of Klavžar, Rajapakse, and Gutman states that η ( G ) ≥ 0 , and by a result of Dobrynin and Gutman η ( 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 η ( G ) = 2 . It is also proved that there is no graph G with the property that η ( G ) = 1 or η ( G ) = 3 . Finally, it is proved that, for a given positive integer k , k ≠ 1 , 3 , there exists a graph G with η ( G ) = k .