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The Schlegel diagram of a closed fullerene is a graph that is planar and 3-regular (or "cubic"; meaning that all vertices have degree 3). A closed fullerene with sphere-like shell must have at least some cycles that are pentagons or heptagons.
In geometry, a Schlegel diagram is a projection of a polytope from into through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in R d − 1 {\textstyle \mathbb {R} ^{d-1}} that, together with the original facet, is combinatorially equivalent to the original polytope.
In graph theory, the term fullerene refers to any 3-regular, planar graph with all faces of size 5 or 6 (including the external face). It follows from Euler's polyhedron formula , V – E + F = 2 (where V , E , F indicate the number of vertices, edges, and faces), that there are exactly 12 pentagons in a fullerene and h = V /2 – 10 hexagons.
Polyfullerene is a basic polymer of the C 60 monomer group, in which fullerene segments are connected via covalent bonds into a polymeric chain without side or bridging groups. They are called intrinsic polymeric fullerenes, or more often all C 60 polymers. Fullerene can be part of a polymer chain in many different ways.
The only two smaller fullerenes are the graph of the regular dodecahedron (a fullerene with 20 vertices) and the graph of the truncated hexagonal trapezohedron (a 24-vertex fullerene), [3] which are the two types of cells in the Weaire–Phelan structure. The 26-fullerene graph has many perfect matchings. One must remove at least five edges ...
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Fullerene or C 60 is soccer-ball-shaped or I h with 12 pentagons and 20 hexagons. According to Euler's theorem these 12 pentagons are required for closure of the carbon network consisting of n hexagons and C 60 is the first stable fullerene because it is the smallest possible to obey this rule.