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

Abstract

We have revisited the Szeged index ( S z ) and the revised Szeged index ( S z ∗ ) , 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 S z / S z ∗ as a measure of bipartivity. In this note we report on some properties of the revised Szeged index and the quotient S z / S z ∗ illustrated on a number of smaller graphs as models of networks.