In Ref we considered the

In Ref. [4], we considered the problem of fullerene growth more elaborately, namely, how to design new fullerenes and their graphs if given a basic elementary graph of a mini-fullerene playing the role of progenitor. We have developed a way of designing different calcium sensing receptor of fullerenes using such approach. As a result, we have found the family of bi-polyfoils: C14, C18, C24, C30, C36; the family of truncated bipyramids: C14, C18, C24 C30, C36; the family of cupola half-fullerenes: C10, C12, C16, C20, C24; and the family of tetra-hexa-cell equator fullerenes: C20, C24, C32, C40, C48. This classification gives not only fullerene symmetry but also connects it with its relatives. It is worth noting that we have obtained a medium size fullerene C36 with D6 symmetry. It was synthesized and separated from arc derived carbon soot at UC, Berkeley [5]. In literature, it is referred to as a 36-atom-carbon cage, but this name is of little value because 15 different isomers are possible [6]. Contrary to this name, our classification determines this fullerene uniquely [4]. However, some midi-fullerenes, e.g., C26, C28[7], were not constructed in Ref. [4], so their graphs were not known. This drawback was excluded in [8] using the graph approach developed in Refs. [2–4]. In Ref. [9], we have suggested a unified approach to drawing axonometric projections for both small and large fullerenes. In the long run, we came to the conclusion that the best way is the dimetric representation whose symmetry coincides with that of a corresponding graph. Then we have carefully studied a dimer mechanism of growing fullerenes, according to which a carbon dimer embeds either into a hexagon or a pentagon of an initial fullerene. This leads to stretching and breaking the covalent bonds which are parallel to the arising tensile forces. In the first case, instead of the hexagon adjoining two pentagons, when the dimer embeds in this hexagon, one obtains two adjacent pentagons adjoining two hexagons. In the second case, when the dimer embeds in the pentagon, two pentagons separated by a square are obtained. In both cases there arises a new atomic configuration and there is a mass increase of two carbon atoms. This process can continue until a new stable configuration is reached. In doing so, we modeled the growth of the first branch of the family of tetra-hexa-cell equator fullerenes beginning with C20 in the range from 20 to 36 together with some of their isomers. We have constructed the axonometric projections and the corresponding graphs for these fullerenes. In this contribution, we consider the direct descendants of the second branch of the family of tetra-hexa-cell equator fullerenes beginning with C24, namely, C2, where n=11–24. Our aim is to study their growth constructing at first their graphs, what is simpler, and then to develop their structure on the basis of the graphs obtained.
Conclusion Any calculations of fullerene energy need input data. For mini-fullerenes (up to C20) the number of possible configurations is not very large, but with midi-fullerenes (C20–C60) a monstrous size of isomers can be obtained. It is clear that there is no big sense in studying all of them, so it is desirable to restrict their number to the most stable. In this respect, geometric modeling is very useful as a first step of computer simulation for further theoretical analysis [10]. As for fullerenes, the geometric modeling is based on the principle “the minimum surface at the maximum volume”. It means that a forming fullerene tends to take the form of a perfect spheroid with equal covalent bonds. We suppose that geometric modeling allows imagining from the very beginning a possible way of growing carbon clusters and thereby to decrease the number of configurations worth for studying. With the help of geometrical modeling, we have considered here the growth of fullerenes through a series of dimer imbedding reactions with initial fullerenes. As a result, axonometric projections together with the corresponding graphs for the second branch of the family of tetra-hexa-cell equator fullerenes including some isomers are constructed in the range from 24 to 48. Some of the graphs were obtained earlier [4] but the majority is given for the first time. The process of growth of fullerenes is studied on the basis of the mechanism, according to which a carbon dimer embeds in a hexagon of an initial fullerene. This leads to stretching and breaking the covalent bonds which are parallel to the arising tensile forces. In this case, instead of the hexagon adjoining two pentagons, two adjacent pentagons adjoining two hexagons are obtained. As a result, there arises a new atomic configuration and there is a mass increase of two carbon atoms. We considered direct descendants of the second branch of the tetra-hexa-cell-equator family beginning with C24, namely, C2, where n=13 – 24.