Combination of distance and symmetry in some molecular graphs

Suppose G is a connected graph or a union of connected graphs and Γ is a subgroup of Aut(G). The modified Wiener index of G with respect to Γ can be defined as follows: W^Γ(G)=|V(G)|2|Γ|∑u∈V(G)∑g∈Γd(u,g(u)).In this article, this graph invariant for the cycle Cn with respect to all subgroups of Aut(Cn) is computed. As consequences, the modified Wiener indices of some molecular graphs like (3, 6)- and (5, 6)-fullerenes with respect to a subgroup of their symmetry groups are computed. Some open questions are also presented.