In the mathematical field of

graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, the 26-fullerene graph is a

polyhedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-c ...

with ''V'' = 26 vertices and ''E'' = 39 edges. Its planar embedding has three hexagonal faces (including the one shown as the external face of the illustration) and twelve pentagonal faces. As a planar graph with only pentagonal and hexagonal faces, meeting in three faces per vertex, this graph is a

fullerene A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...

. The existence of this fullerene has been known since at least 1968.

Properties

The 26-fullerene graph has

D_

prismatic symmetry In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2). Types Ther ...

, the same group of symmetries as the

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. ...

. This symmetry group has 12 elements; it has six symmetries that arbitrarily permute the three hexagonal faces of the graph and preserve the

orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building desi ...

of its planar embedding, and another six orientation-reversing symmetries. The number of fullerenes with a given even number of vertices grows quickly in the number of vertices; 26 is the largest number of vertices for which the fullerene structure is unique. The only two smaller fullerenes are the graph of the

regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, ...

(a fullerene with 20 vertices) and the graph of the truncated hexagonal trapezohedron (a 24-vertex fullerene), which are the two types of cells in the

Weaire–Phelan structure In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...

. The 26-fullerene graph has many

perfect matching In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph , a perfect matching in is a subset of edge set , such that every vertex in the vertex set is adjacent to exactly ...

s. One must remove at least five edges from the graph in order to obtain a subgraph that has exactly one perfect matching. This is a unique property of this graph among fullerenes in the sense that, for every other number of vertices of a fullerene, there exists at least one fullerene from which one can remove four edges to obtain a subgraph with a unique perfect matching. The vertices of the 26-fullerene graph can be labeled with sequences of 12

bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...

s, in such a way that distance in the graph equals half of the

Hamming distance In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chang ...

between these bitvectors. This can also be interpreted as an

isometric embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...

from the graph into a 12-dimensional

taxicab geometry A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian ...

. The 26-fullerene graph is one of only five fullerenes with such an embedding.. For the embedding, see Figure 5.3, p. 52. 26fullerene distance labels

In popular culture

In 2009, ''

The New York Times ''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...

'' published a puzzle involving

Hamiltonian path In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

s in this graph, taking advantage of the correspondence between its 26 vertices and the 26 letters of the English alphabet.

References

{{reflist Individual graphs Planar graphs