1. General fullerenes, graphs, symmetry and isomersFrom the tetravalence of C result four possible vertex situations,
that could be denoted as 31, 22, 211 and 1111 (Fig.
1a). The situation 31 could be obtained by adding two C atoms between
any two others connected by a single bond, and situation 22 by adding a
C atom between any two others connected by a double bond (Fig.
1b). Therefore, we could restrict our consideration to the remaining
two non-trivial cases: 211 and 1111. Working in opposite sense, we could
always delete 31 or 22 vertices, and obtain a reduced 4-regular graph,
where in each vertex occurs at most one double bond (digon), that could
be denoted by colored (bold) edge (Fig.
1a). First, we could consider all 4-regular graphs on a sphere, from
which non-trivial in the sense of derivation are only reduced ones. In
the knot theory, 4-regular graphs on a sphere with all vertices of the
type 1111 are known as "basic polyhedra" [1,2,
3,4],
and that with at least one vertex with a digon as "generating knots or
links" [4]. From the chemical reasons,
the vertices of the type 1111 are only theoretically acceptable. If all
the vertices of such 4-regular graph are of the type 211, such graph we
will be called a S_{5} of order 120 [5],
but for C_{80} with the same G, G' is always a proper
subgroup of G, and its chemical symmetry is lower than the geometrical.
Hence, after C_{60}, the first fullerene with G=G'=
[3,5]=I=_{h}S_{5}
will be C_{180}, then C_{240}, etc.
Working with general fullerenes without any restriction
for the number of edges of their faces, the first basic polyhedron from
which we could derive them (after the trivial 1*) will be the regular octahedron
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