Use of the Szeged index and the revised Szeged index for measuring network bipartivity

We have revisited the Szeged index (Sz)(Sz) and the revised Szeged index (Sz∗)(Sz∗), both of which represent a generalization of the Wiener number to cyclic structures. Unexpectedly we found that the quotient of the two indices offers a novel measure for characterization of the degree of bipartivity of networks, that is, offers a measure of the departure of a network, or a graph, from bipartite networks or bipartite graphs, respectively. This is because the two indices assume the same values for bipartite graphs and different values for non-bipartite graphs. We have proposed therefore the quotient Sz/Sz∗Sz/Sz∗ as a measure of bipartivity. In this note we report on some properties of the revised Szeged index and the quotient Sz/Sz∗Sz/Sz∗ illustrated on a number of smaller graphs as models of networks.