Generate different types of fullerenes.
A fullerene is a plane graph whose faces are all pentagons and hexagons.
This is a website interface to the program fullgen, written by Gunnar Brinkmann.
The default graph format are adjacency lists, where the neighbors of each vertex are listed in clockwise (cw) order around the vertex in the embedding of the graph.

Number $n$ of vertices

(even, between 20 and 80/120)

Pentagons isolated (IPR)

Filter by symmetries

Graph format

Output format

Output
numbering
graphics

Object info

The number of vertices of any fullerene must be even.
Moreover, by Euler's formula, there must always be exactly 12 pentagons.
For any fullerene with $n$ vertices, the number of hexagons is $n/2-10$, and the number of edges is $3n/2$.
The dodecahedron shown below is the smallest fullerene with 20 vertices.
Its adjacency list representation is 1[10 11 2] 2[1 3 16] 3[2 15 4] 4[3 5 17] 5[4 14 6] 6[5 7 18] 7[6 13 8] 8[7 9 19] 9[8 12 10] 10[9 1 20] 11[1 12 15] 12[11 9 13] 13[12 7 14] 14[13 5 15] 15[14 3 11] 16[2 17 20] 17[16 4 18] 18[17 6 19] 19[18 8 20] 20[19 10 16].

We say that the pentagons in a fullerene are isolated, if no two of them share an edge.
Such fullerenes are called IPR fullerenes (IPR stands for 'isolated pentagon rule').
The smallest IPR fullerene is the Buckminsterfullerene or buckyball and has 60 vertices.
Please refer to a chemical dictionary for the meaning of the 28 different symmetry groups.

Enumeration (OEIS)

The number of fullerenes is given by OEIS A007894, and the number of IPR fullerenes is given by OEIS A046880.

[BD97] G. Brinkmann and A. W. M. Dress. A constructive enumeration of fullerenes. J. Algorithms, 23(2):345–358, 1997.

[BGM12] G. Brinkmann, J. Goedgebeur, and B. D. McKay. The generation of fullerenes. Journal of Chemical Information and Modeling, 52(11):2910–2918, 2012.