Computation of certain topological indices of nanotubes

Among topological connectivity indices, the atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are of vital importance. The ABC index is defined as: ABC(G) = Σ uvεE(G) √ (dG(u)+dG(v)-2)/(dG(u)dG(v)), while GA(G) index also defined as follows: GA(G) = Σ uvεE(G)((2 √ dG(u)dG(v))/(dG(u)+dG(v))), where dG(u) denotes the degree of vertex u ε V (G). Recently, the fourth version of ABC index is proposed by Ghorbani et al. defined as follows: ABC₄(G) = Σ uvεE(G) √ (δG(u)+δG(v)-2)/(δG(u)δG(v)). The fifth version of GA index is introduced by Graovac et al. defined as follows: GA₅(G) = Σ uvεE(G)((2 √ δG(u)δG(v))/(δG(u)+δG(v))), where δu = Σ uvεE(G) dG(v). In this paper, we study the fourth atom-bond connectivity index ABC₄ and fifth geometric-arithmetic index GA₅ and give close formulae of these indices for HAC₅C₇[p,q], HAC₅C₆C₇[p,q] and TUC₄C₈(R)[p,q] nanotubes and their corresponding nanotori. We also give a characterization of k-regular graphs with respect to their fifth geometric-arithmetic index.