3.3.2 5/6 fullerenesAmong general fullerenes we can distinguish the class consisting of 5/6 fullerenes having only pentagonal and hexagonal faces. If n_{5} is the number of pentagons, and n_{6} the number of hexagons, from the relationship 3v = 2e and the Euler theorem it follows directly that n_{5} = 12. Hence, the first 5/6 fullerene will be C_{20} with n_{6} = 0, the regular dodecahedron {5,3}. It has two nonisomorphic edgecolorings, resulting in two chemically different isomers of the same geometrical dodecahedral form. The first basic polyhedron generating 5/6 fullerenes is the one with v = 10 vertices. For v = 10, there are three basic polyhedra, but only 10^{*} and 10^{**} generate 5/6 fullerenes, each only one of them (b,c). On the other hand, they generate, respectively, 78 and 288 general fullerenes. Their number is equal to the number of source links derived from the basic polyhedra 10^{*} and 10^{**} in the Section 2.5. There are two mutually dual methods for the derivation of fullerenes:
This provides a double check of the obtained results. The duality of these methods is illustrated by the example of two C_{20} chemical isomers both derived from the same geometrical dodecahedral form in which G = [3,5] = I_{h} = S_{5} of order 120. However, for the first G' = D_{5d} = [2+,10] = D_{5}×C_{2} of order 20, and for the other G' = [2,2]+ = D_{2} of order 4 (a,b). In this case, the symmetry of chemical isomers derived by the vertex bifurcation is preserved from their generating basic polyhedra (b). For the enumeration of general fullerenes (i.e., source links derived from basic polyhedra in the Section 2.5) we used the Polya Enumeration Theorem (PET), applied to basic polyhedra knowing their automorphism groups (see the Section 2.5), but its application to 5/6 fullerenes is not possible. The same restriction holds for the other derivation method, because of the condition that in every vertex exactly one edge of a 3regular graph must be colored. The 3valent graphs with n < 13 vertices and their edgecolorings producing 4valent graphs are considered by A.Yu. Vesnin (1991). Proceeding in the same way, it is possible to prove that 5/6 fullerenes with 22 atoms cannot exist, and there are seven 5/6 fullerenes C_{24} with the same geometrical form and G = D_{6d} = [2+,12] = D_{12}. Often, even knowing chemical symmetry groups G' is not sufficient for distinguishing chemical isomers. Some results from knot theory: the polynomial invariants of knot and link projections (see the Section 2.8) can solve this problem. Every 4valent planar graph can be transformed into the projection of an alternating KL (and vice versa), and the correspondence between such alternating KL diagrams and 4valent graphs is 11 (up to chirality).
