  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, in 2018 the connection between the
Graovac-Pisanski index and the melting points of some hydrocarbon molecules was
established. 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. These results are applied to some important
families of hydrocarbon molecules. 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.
