On the differences between Szeged and Wiener indices of graphs

Let G be a connected graph and η(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 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.

► We prove that there is no graph G such that η(G)=Sz(G)−W(G) is equal to 1 or 3. ► We prove that η(G)=2 if and only if G is a member of a class K2. ► For every non-negative integer n≠1 or 3, there exists a graph G such that η(G)=n. ► If a block B of G contains an isometric cycle of length n≥4, then η(G)≥n.