The Graovac-Pisanski index of a connected bipartite graph is an integer number

Matevž Črepnjak, Martin Knor, Niko Tratnik, Petra Žigert Pleteršek

Published 2017-09-13Version 1

The Graovac-Pisanski index, also called the modified Wiener index, was introduced in 1991 and represents an extension of the original Wiener index, because it considers beside the distances in a graph also its symmetries. Similarly as Wiener in 1947 showed the correlation of the Wiener indices of the alkane series with the boiling points, \v{C}repnjak, Tratnik, and \v{Z}igert Pleter\v{s}ek recently found out that the Graovac-Pisanski index is correlated with the melting points of some hydrocarbon molecules. The examples showed that for all the considered molecular graphs the Graovac-Pisanski index is an integer number. In this paper, we prove that the Graovac-Pisanski index of any connected bipartite graph as well as of any connected graph on an even number of vertices is an integer number. By using a computer programme, the graphs with a non-integer Graovac-Pisanski index on at most nine vertices are counted. Finally, an infinite class of unicyclic graphs with a non-integer Graovac-Pisanski index is described.