A *nut graph* is a graph of at least 2 vertices whose adjacency matrix has nullity 1 (i.e., rank n-1 where n is the order of the graph) and for which the non-trivial kernel vector does not contain a zero.

The graph lists are currently only available in 'graph6' format. The larger files are compressed with gzip. Please refer to [1] for more information on how these nut graphs were obtained.

The following lists are available:

Vertices | girth ≥ 3 | girth ≥ 4 | girth ≥ 5 | girth ≥ 6 | girth ≥ 7 | girth ≥ 8 | girth ≥ 9 |
---|---|---|---|---|---|---|---|

0-6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

7 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |

9 | 560 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 12551 | 0 | 0 | 0 | 0 | 0 | 0 |

11 | 2060490 | 16 | 2 | 0 | 0 | 0 | 0 |

12 | 208147869 | 22 | 2 | 0 | 0 | 0 | 0 |

13 | 96477266994 | 3909 | 20 | 0 | 0 | 0 | 0 |

14 | ? | 19029 | 35 | 0 | 0 | 0 | 0 |

15 | ? | 3641664 | 1027 | 6 | 1 | 0 | 0 |

16 | ? | 48040271 | 2415 | 5 | 0 | 0 | 0 |

17 | ? | ? | 88973 | 155 | 1 | 0 | 0 |

18 | ? | ? | 341499 | 139 | 0 | 0 | 0 |

19 | ? | ? | 14161780 | 6146 | 37 | 1 | 1 |

20 | ? | ? | 82020028 | 6668 | 8 | 0 | 0 |

The following table contains the counts of chemical nut graphs. A *chemical graph* is a connected graph with maximum degree at most 3.

Vertices | girth ≥ 3 | girth ≥ 4 | girth ≥ 5 | girth ≥ 6 | girth ≥ 7 | girth ≥ 8 | girth ≥ 9 |
---|---|---|---|---|---|---|---|

0-8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

9 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

11 | 8 | 1 | 0 | 0 | 0 | 0 | 0 |

12 | 9 | 2 | 1 | 0 | 0 | 0 | 0 |

13 | 27 | 4 | 2 | 0 | 0 | 0 | 0 |

14 | 23 | 1 | 1 | 0 | 0 | 0 | 0 |

15 | 414 | 76 | 25 | 0 | 0 | 0 | 0 |

16 | 389 | 50 | 14 | 1 | 0 | 0 | 0 |

17 | 7941 | 1788 | 424 | 16 | 1 | 0 | 0 |

18 | 8009 | 1267 | 188 | 6 | 0 | 0 | 0 |

19 | 67970 | 15251 | 5583 | 308 | 10 | 0 | 0 |

20 | 51837 | 6576 | 2764 | 136 | 1 | 0 | 0 |

21 | 1326529 | 331301 | 116524 | 6525 | 90 | 6 | 1 |

22 | 1372438 | 231356 | 73941 | 2964 | 27 | 0 | 0 |

The following table contains the counts of nut graphs among cubic polyhedra. A *polyhedron* is a 3-connected simple planar graph. For more information on cubic polyhedra, see the Planar graphs page.

Vertices | Cubic polyhedra | Nut graphs |
---|---|---|

4 | 1 | 0 |

6 | 1 | 0 |

8 | 2 | 0 |

10 | 5 | 0 |

12 | 14 | 2 |

14 | 50 | 0 |

16 | 233 | 0 |

18 | 1249 | 285 |

20 | 7595 | 0 |

22 | 49566 | 0 |

24 | 339722 | 62043 |

24 | 339722 | 62043 |

26 | 2406841 | 4 |

28 | 17490241 | 316 |

30 | 129664753 | 16892864 |

32 | 977526957 | 3676 |

34 | 7475907149 | 447790 |

The following table contains the counts of nut graphs among fullerenes up to 250 vertices. For those orders up to 250 where no count is listed, the implication is that there is no nut fullerene of that order. Fullerenes are cubic plane graphs where all faces are pentagons or hexagons. For more information on fullerenes, see the Fullerene page. The fullerene lists are currently only available in 'planar_code' format.

Vertices | Nut fullerenes |
---|---|

36 | 1 |

42 | 1 |

44 | 1 |

48 | 2 |

52 | 2 |

60 | 6 |

72 | 2 |

82 | 1 |

84 | 8 |

96 | 5 |

108 | 7 |

120 | 5 |

132 | 14 |

144 | 6 |

156 | 11 |

160 | 1 |

168 | 11 |

180 | 16 |

192 | 8 |

204 | 19 |

216 | 9 |

228 | 21 |

240 | 16 |

The following table contains the counts of all *d*-regular nut graphs with a specified degree *d*.
Please refer to [2] and [3] for more information on how these nut graphs were obtained.

Vertices | Degree 3 | Degree 4 | Degree 5 | Degree 6 | Degree 7 | Degree 8 |
---|---|---|---|---|---|---|

0-7 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 0 | 1 | 0 | 0 | 0 | 0 |

9 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 0 | 12 | 9 | 0 | 0 | 0 |

11 | 0 | 0 | 0 | 0 | 0 | 0 |

12 | 9 | 269 | 4 | 1964 | 3 | 24 |

13 | 0 | 0 | 0 | 79 | 0 | 0 |

14 | 0 | 15633 | 25 | 1872 | 5168453 | 424088 |

15 | 0 | 1 | 0 | ? | 0 | ? |

16 | 0 | ? | 13530 | ? | ? | ? |

17 | 0 | ? | 0 | ? | 0 | ? |

18 | 5541 | ? | 665456900 | ? | ? | ? |

19 | 0 | ? | 0 | ? | 0 | ? |

20 | 5 | ? | ? | ? | ? | ? |

21 | 0 | ? | 0 | ? | 0 | ? |

22 | 71 | ? | ? | ? | ? | ? |

[1] K. Coolsaet, P.W. Fowler and J. Goedgebeur, Generation and properties of of nut graphs, MATCH Commun. Math. Comput. Chem., 80(2):423-444, 2018.

[2] J.B. Gauci, T. Pisanski and I. Sciriha, Existence of regular nut graphs and the Fowler extension, arXiv preprint arXiv:1904.02229, 2019.

[3] P.W. Fowler, J.B. Gauci, J. Goedgebeur, T. Pisanski and I. Sciriha, Existence of regular nut graphs for degree at most 11, to appear in Discussiones Mathematicae Graph Theory, 17 pages, 2019.