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5. Nanotubes, conical and biconical fullerenes and their symmetry

For denoting different categories of symmetry groups, we will use Bohm symbols [16]. In a symbol Gnst, the first subscript n represents the maximal dimension of space in which the transformations of the symmetry group act, while the following subscripts st represent the maximal dimensions of subspaces remaining invariant under the action of transformations of the symmetry group, that are properly included in each other. With regard to their symmetry, general fullerenes belong to the category of point groups G30. The category G30 consists of seven polyhedral symmetry groups without invariant planes or lines: [3,3] or Td, [3,3]+ or T, [3,4] or Oh, [3,4]+ or O, [3+,4] or Th, [3,5] or Ih, [3,5]+ or I, and from seven infinite classes of point symmetry groups with the invariant plane (and the line perpendicular to it in the invariant point): [q] or Cqv, [q]+ or Cq, [2+,2q+] or S2q, [2,q+] or Cqh, [2,q]+ or Dq, [2+,2q] or Dqd, [2,q] or Dqh, belonging to the subcategory G320 [5]. For the groups of the subcategory G320, in the case of rotations of order q>2, the invariant line (i.e. the rotation axis) may contain 0,1 or 2 vertices of a general fullerene. According to this, among all general fullerenes with a geometrical symmetry group G belonging to G320, from the topological point of view we could distinguish, respectively, cylindrical fullerenes (nanotubes), conical and biconical ones. 

We could simply conclude that for polyhedral 5/6 fullerenes G could be only [3,3] (Td ), [3,3]+ (T), [3,5] (Ih), [3,5]+ (I), because of their topological structure (n5=12), incompatible with the octahedral symmetry group [3,4] (Oh) or its polyhedral subgroups. In the case of nanotubes (or cylindrical fullerenes) we have infinite classes of 5/6 fullerenes with the geometrical symmetry group [2,q] (Dqh) and [2+,2q] (Dqd), and the same chemical symmetry. The first infinite first class of cylindrical nanotubes with G=G'=D5h we obtain from a cylindrical 3/4/5 4-regular graph with two pentagonal bases, 10 triangular and 5(2k+1) quadrilateral faces (k=0,1,2,) and with the same symmetry group (Fig. 11). By the vertex bifurcation preserving its symmetry, we obtain the infinite class of nanotubes C30, C50, C70,, with C70 as the first of them satisfying IPR. Certainly, the geometrical structure of C70 admits different edge colorings (i.e. chemical isomers). Starting from any two of them (Fig. 12) by "collapsing" (the inverse of "vertex bifurcation", i.e. by deleting digons) we could obtain different generating 4-regular graphs. This example of two different C70 isomers, with the same geometrical structure, and with the same G and G', shows that for the exact recognition of fullerene isomers we need to know more than their geometrical and chemical symmetry (see Part 3). In the same way, from 4-regular graphs with two hexagonal bases, 12 triangular and 6(2k+1) quadrilateral faces (k=0,1,2,) we obtain the infinite class of fullerenes C36, C60 C84, with the symmetry group G=G'=D6h (Fig. 13). 
  

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Figure 11. 
  
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Figure 12. 
  
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Figure 13. 

The next symmetry groups [2+,2q] (Dqd) with q=5,6 we obtain in the same way, from 4-regular graphs with q-gonal bases, 2q triangular and 2kq quadrilateral faces (k=1,2, for q=5; k=0,1,2, for q=6) (Fig. 14). As the limiting case, for q=5 and k=0, we obtain C20 with the icosahedral symmetry group G, but with G'=D5d, that could be used as the "brick" for the complete class of nanotubes C40, C60,C80, with G= D5d, where all of them could be obtained from C20 by "gluing" the pentagonal bases (Fig. 15). In the same way, fullerene C24 obtained for q=6 and k=0 could be used as the building block for the nanotubes C48, C72,C96, The geometrical structure of nanotube class with G=Dqd (q=5,6) permits the edge coloring preserving the symmetry, so there always exist their isomers with G=G'. 
  

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Figure 14. 
  
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Figure 15. 

If the 3-rotation axis contains the opposite vertices of a fullerene, we have biconical fullerenes (e.g. C26, C56) with G=D3h, G=D3d, respectively (Fig. 16). Certainly, after the edge coloring, their symmetry must be disturbed, and for them G' is always a proper subgroup of G. For example, for C26 (Fig. 16), G=D3h, G'=C2v.
  

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Figure 16. 

Proceeding in the same way, it is possible to find or construct fullerene representatives of other symmetry groups from the category G320: biconical C32 with G=D3, biconical C38 or conical C34 with G=C3v, conical C46 with G=C3 [17], or the infinite class of cylindrical fullerenes C42, C48, C54 with G=D3 (Fig. 17). In general, after edge coloring of their 3-regular graphs, symmetry could not be preserved in all conical or biconical fullerenes mentioned, so their geometrical symmetry is always higher than the chemical. 
  

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Figure 17. 

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