The Graovac-Pisanski index of connected bipartite graphs with applications to hydrocarbon molecules

Abstract 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.

[1]  F. Koorepazan-Moftakhar,et al.  Combination of distance and symmetry in some molecular graphs , 2016, Appl. Math. Comput..

[2]  Aleksandra Tepeh,et al.  Trees with the maximal value of Graovac-Pisanski index , 2019, Appl. Math. Comput..

[3]  Riste Skrekovski,et al.  Mathematical aspects of fullerenes , 2016, Ars Math. Contemp..

[4]  Roberto Todeschini,et al.  Handbook of Molecular Descriptors , 2002 .

[5]  Niko Tratnik,et al.  The Graovac–Pisanski index of armchair tubulenes , 2017, Journal of Mathematical Chemistry.

[6]  Niko Tratnik,et al.  Predicting melting points of hydrocarbons by the Graovac-Pisanski index , 2017, 1709.01832.

[7]  Sandi Klavzar,et al.  Modified Wiener index via canonical metric representation, and some fullerene patches , 2016, Ars Math. Contemp..

[8]  Riste Škrekovski,et al.  On Diameter of Nanotubical Fullerene Graphs , 2015 .

[9]  Rodolfo Pinal,et al.  Effect of molecular symmetry on melting temperature and solubility. , 2004, Organic & biomolecular chemistry.

[10]  Ali Reza Ashrafi,et al.  Graovac–Pisanski index of fullerenes and fullerene–like molecules , 2016 .

[11]  Ante Graovac,et al.  On the Wiener index of a graph , 1991 .

[12]  Ali Reza Ashrafi,et al.  The modified Wiener index of some graph operations , 2016, Ars Math. Contemp..

[13]  Modjtaba Ghorbani,et al.  On the Graovac-Pisanski index , 2017 .

[14]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[15]  Ali Reza Ashrafi,et al.  Distance, Symmetry, and Topology in Carbon Nanomaterials , 2016 .

[16]  A. Ashrafi,et al.  Symmetry and PI Polynomials of C50+10n Fullerenes , 2014 .

[17]  Ali Reza Ashrafi,et al.  Topological Symmetry of Nanostructures , 2015 .

[18]  N. Tratnik.  The Graovac–Pisanski index of zig-zag tubulenes and the generalized cut method , 2017, Journal of Mathematical Chemistry.