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Encoding fullerenes and geodesic domes
 SIAM. J. Discrete Math
, 2004
"... Abstract. Coxeter’s classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with label ..."
Abstract

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Abstract. Coxeter’s classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with labeled angles and edges. In the case of the Coxeter geodesic domes, the plane graph is the icosahedron, all angles are labeled one, and all edges are labeled by the same pair of integers (p, q). Edges with these “Coxeter coordinates ” correspond to straight line segments joining two vertices of Λ, the regular triangular tessellation of the plane, and the faces of the icosahedron are filled in with equilateral triangles from Λ whose sides have coordinates (p, q). We describe the construction of the signature for any geodesic dome. In turn, we describe how each geodesic dome may be reconstructed from its signature: the angle and edge labels around each face of the signature identify that face with a polygonal region of Λ and, when the faces are filled by the corresponding regions, the geodesic dome is reconstituted. The signature of a fullerene is the signature of its dual. For each fullerene, the separation of its pentagons, the numbers of its vertices, faces, and edges, and its symmetry structure are easily computed directly from its signature. Also, it is easy to identify nanotubes by their signatures.
Extremal fullerene graphs with the maximum Clar number
, 801
"... A fullerene graph is a cubic 3connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The ma ..."
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Cited by 1 (1 self)
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A fullerene graph is a cubic 3connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar number is no more than ⌊n−12 6 ⌋. Many fullerenes with experimental evidence attain the upper bound, for instance, C60 and C70. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal n−12 6. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C60, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.