Nut graphs

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:

Nut graphs

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 ? 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

Chemical nut graphs

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

Nut graphs among cubic polyhedra

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

Nut fullerenes

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

References

[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.