Fullerenes as Tilings of Surfaces

Abstract

If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Klein-bottle fullerenes, none. Klein-bottle and elliptic fullerenes are the antipodal quotients of centrosymmetric toroidal and spherical fullerenes, respectively. Extensions to infinite systems (plane fullerenes, cylindrical fullerenes, and space fullerenes) are indicated. Eigenvalue spectra of all four classes of finite fullerenes, are reviewed. Leapfrog fullerenes have equal numbers of positive and negative eigenvalues, with 0, 0, 2, or 4 eigenvalues zero for spherical, elliptic, Klein-bottle, and toroidal cases, respectively.

Some Algebraic Polynomials and Topological Indices of Generalized Prism and Toroidal Polyhex Networks

“…Two vertices are adjacent if there is a bond between corresponding atoms. In [25], authors considered fullerene's extension to other closed surfaces and showed that only four surfaces, sphere, torus, Klein bottle and projective (elliptic) plane, are possible. The spherical and elliptic fullerenes have 12 and 6 pentagons respectively.…”

Design and realization of macroscopic grid architectures mimicking carbon molecules

“…A fullerene can also be seen as a trivalent graph with hexagonal and pentagonal faces [8][9][10]. Fullerenes, and nanotubes in particular, can be obtained by rolling a sheet of graphite.…”

k-resonance in Toroidal Polyhexes*

“…Its molecular graph is a finite trivalent graph embedded on the surface of a sphere with only hexagonal and (exactly 12) pentagonal faces. Deza et al [4] considered fullerene's extension to other closed surfaces and showed that only four surfaces are possible: sphere, torus, Klein bottle and projective plane. Unlike spherical fullerenes, toroidal and Klein bottle's fullerenes have been regarded as tessellations of entire hexagons on their surfaces since they must contain no pentagons [4,11].…”

“…Deza et al [4] considered fullerene's extension to other closed surfaces and showed that only four surfaces are possible: sphere, torus, Klein bottle and projective plane. Unlike spherical fullerenes, toroidal and Klein bottle's fullerenes have been regarded as tessellations of entire hexagons on their surfaces since they must contain no pentagons [4,11]. For the theoretical consideration and detailed classifications of hexagonal tilings (dually, triangulations) on the torus and the Klein bottle, see [17,22].…”