# The classification of ??-embeddable fullerenes

Abstract (Summary)

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Sergey Shpectorov, Advisor
In Chemistry, fullerenes are molecules composed entirely of carbon atoms, in the form
of a hollow sphere, ellipsoid or tube, such that each atom is bonded with three other atoms
and the atoms form pentagonal or hexagonal rings. The spherical fullerenes motivated the
related mathematical concept: a fullerene graph is a trivalent plane graph such that all faces
are pentagons and hexagons.
The goal of this research is to prove the conjecture that there are exactly five ?1embeddable
fullerenes. These are known to be the following fullerenes: F20(Ih), F26(D3h),
F40(Td), F44(T ) and F80(Ih) (where the group of symmetry is given in parentheses for each
fullerene). We proceed in proving this result by looking at the minimal distance between
the pentagonal faces of the fullerene. In the cases when the minimal distance between pentagons
is greater than two we obtain a contradiction, which leads us to conclude that in an
?1-embeddable fullerene there must exist at least two pentagons that either are adjacent or
have a common hexagonal neighbor. For the latter cases we show that the only possibilities
are the five fullerenes listed above.
Bibliographical Information:

Advisor:

School:Bowling Green State University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:fullerenes

ISBN:

Date of Publication: