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

  • Authors:

  • Toma Pisanski;Milan Randić

  • Affiliations:

  • Institute of Mathematics, Physics & Mechanics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia and University of Primorska, Primorska Institute of Natural Sciences and Technology, ...;National Institute of Chemistry, Hajdrihova 19, 1000 Ljubljana, Slovenia

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

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