3.3.1  General fullerenes, graphs, symmetry and isomers

From the tetravalence of carbon we get four possible vertex situations shown in the following figure. These are denoted 31, 22, 211 and 1111. The situation 31 is obtained by adding two carbon atoms between any two others connected by a single bond, and situation 22 by adding a carbon atom between any two others connected by a double bond (b). Therefore, we can restrict our consideration to the remaining two non-trivial cases: 211 and 1111. Working in the opposite sense, we can always delete 31 or 22 vertices and obtain a reduced 4-valent graph, where in each vertex occurs at most one double bond (bigon), which can be denoted by colored (bold) edge (a). First, we can consider all 4-valent graphs on a sphere. For chemical reasons, vertices of type 1111 are only theoretically acceptable. In knot theory, 4-valent graphs on a sphere with all vertices of the type 1111 are basic polyhedra. If all the vertices of such 4-valent graph are of the type 211, such graph we will be called a general fullerene. Every general fullerene can be derived from a basic polyhedron by a vertex bifurcation, this means, by replacing its vertices by bigons, with two possibilities for their placement (2 and 2 0) (c). In knot theory, general fullerenes are polyhedral source links. To every general fullerene we can assign an edge-colored 3-valent graph (with bold edges denoting bigons), unique up to isomorphism. In this way, we have two complementary ways for the derivation of general fullerenes: the vertex bifurcation method applied to basic polyhedra, and the edge-coloring method applied to 3-valent graphs, where each vertex has exactly one colored edge. For every general fullerene we can describe its geometrical structure (i.e., the positions of C atoms) by a non-colored 3-valent graph, while its chemical structure (i.e., positions of C atoms and their double bonds) is described by the corresponding edge-colored 3-valent graph. Likewise, for every general fullerene we can distinguish two possible symmetry groups: a symmetry group G corresponding to the geometrical structure and its subgroup G' corresponding to the chemical structure. In the same sense, we will distinguish geometrical and chemical isomers. For example, for C60, G = G' = [3,5] = Ih = S5 of order 120 (Coxeter and Moser, 1980), but for C80 with the same G, G' is always a proper subgroup of G, and its chemical symmetry is lower than the geometrical. Hence, the first fullerene with G = G' = [3,5] = Ih = S5 after C60 is C180, then C240, etc

Working with general fullerenes without restrictions on the number of edges of their faces, the first basic polyhedron from which they can be derived (after the trivial 1*) will be the regular octahedron {3,4} or 6*, that gives 7 general fullerenes. From the basic polyhedron 8* with v = 8 we derive 30, and from the basic polyhedron 9* we obtain 4 general fullerenes, etc. In fact, this list of general fullerenes derived from basic polyhedra is identical with the list of source links derived from basic polyhedra (see the Section 2.5). 

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