Modified Wiener index via canonical metric representation, and some fullerene patches

A variation of the classical Wiener index, the modified Wiener index, that was introduced in 1991 by Graovac and Pisanski , takes into account the symmetries of a given graph. In this paper it is proved that the computation of the modified Wiener index of a graph G  can be reduced to the computation of the Wiener indices of the appropriately weighted quotient graphs of the canonical metric representation of G . The computation simplifies in the case when G  is a partial cube. The method developed is applied to two infinite families of fullerene patches.

[1]  Jack E. Graver,et al.  Fullerene patches II , 2014, Ars Math. Contemp..

[2]  Christina M. Graves,et al.  Fullerene patches I , 2010, Ars Math. Contemp..

[3]  Ali Reza Ashrafi and Zohreh Mohammad-Abadi.  On Wiener Index of One-Heptagonal Nanocone , 2012 .

[4]  Nenad Trinajstić,et al.  Wiener index revisited , 2001 .

[5]  Sandi Klavžar,et al.  The Edge – Wiener Index of Benzenoid Systems in Linear Time , 2015 .

[6]  Bolian Liu,et al.  A Survey on Recent Results of Variable Wiener Index , 2013 .

[7]  Sandi Klavzar,et al.  Wiener index in weighted graphs via unification of Θ∗Θ∗-classes , 2014, Eur. J. Comb..

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

[9]  I. Gutman,et al.  A class of modified Wiener indices , 2004 .

[10]  W. Imrich,et al.  Handbook of Product Graphs, Second Edition , 2011 .

[11]  Sandi Klav.  On the canonical metric representation, average distance, and partial Hamming graphs , 2004 .

[12]  Sandi Klavzar and Mohammad J. Nadjafi-Arani.  Cut Method: Update on Recent Developments and Equivalence of Independent Approaches , 2014 .

[13]  R. Graham,et al.  On isometric embeddings of graphs , 1985 .

[14]  Ante Graovac,et al.  On Modified Wiener Indices of Thorn Graphs , 2004 .

[15]  Sandi Klavzar,et al.  Wiener Number of Vertex-weighted Graphs and a Chemical Application , 1997, Discret. Appl. Math..

[16]  S. Klavžar.  On the canonical metric representation of graphs and partial Hamming graphs , 2004 .

[17]  Ali Reza Ashrafi,et al.  Distance Under Symmetry: (3,6)-Fullerenes , 2016 .

[18]  Harishchandra S. Ramane,et al.  ON THE WIENER INDEX OF A GRAPH , 2012 .