Nanoscale structures of carbon display an attractive variation of structural characteristics, and many useful forms have been synthesized and identified. One of these structures is the carbon nanotubes. Carbon nanotubes are either single-wall (SWCNT) or multi-wall; the former attract more attention due to their unique electronic, optical and spectroscopic properties. The electrochemical properties of the fullerene C-60 have been studied previously. The accommodation of C60 inside the SWCNT leads to significant periodic modi. cations of the electronic states of SWCNT. One of the main recognized structures of nanotubes is the (5, 5) single-wall tube (SWCN). A topological index is a mathematical invariant of a chemical graph, which shows a significant correlation with some chemical or physical property. They have been successfully used to construct effective and useful mathematical methods for finding good relationships between structural data and the properties of these materials. To establish a good structural relationship between the structure of molecules C60 and [SWCN(5,5)-Armchair-CnH20] (n = 20-190) 1-18, the molecular degree of unsaturation (D-U) was used as one of the useful numerical and structural electrochemical properties of unsaturated compounds. In this study, the relationship between this index and electron affinity; the reduction potential (E-Red) of [SWCN(5,5)-Armchair-CnH20] 1-18 and 19-29; and the free energy of electron transfer (Delta G(et)) as assessed using the Rehm-Weller equation between 1-18 and fullerene C-60 as C-60@[SWCN(5,5)-Armchair-CnH20] 30-47 and 48-58 complexes, are presented. (C) 2008 Elsevier B. V. All rights reserved.