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
