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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.

Nut graphs
VerticesGirth ≥ 3Girth ≥ 4Girth ≥ 5Girth ≥ 6Girth ≥ 7Girth ≥ 8Girth ≥ 9
0-60000000
73000000
813000000
9560000000
1012551000000
1120604901620000
122081478692220000
13964772669943909200000
14?19029350000
15?364166410276100
16?4804027124155000
17??88973155100
18??341499139000
19??1416178061463711
20??820200286668800
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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.

VerticesGirth ≥ 3Girth ≥ 4Girth ≥ 5Girth ≥ 6Girth ≥ 7Girth ≥ 8Girth ≥ 9
0-80000000
91000000
100000000
118100000
129210000
1327420000
1423110000
1541476250000
1638950141000
177941178842416100
18800912671886000
19679701525155833081000
205183765762764136100
21132652933130111652465259061
2213724382313567394129642700
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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.

VerticesCubic polyhedraNut graphs
410
610
820
1050
12142
14500
162330
181249285
2075950
22495660
2433972262043
2624068414
2817490241316
3012966475316892864
329775269573676
347475907149447790
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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.

VerticesNut fullerenes
361
421
441
482
522
606
722
821
848
965
1087
1205
13214
1446
15611
1601
16811
18016
1928
20419
2169
22821
24016
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Regular nut graphs

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.

VerticesDegree 3Degree 4Degree 5Degree 6Degree 7Degree 8
0-7000000
8010000
9000000
100129000
11000000
12926941964324
130007900
140156332518725168453424088
150101536656800430798
160134871913530???
17042210?0?
18554175138153665456900???
190?0?0?
205?????
210?0?0?
2271?????
230?0?0?
2413184574?????
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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.

[2] J.B. Gauci, T. Pisanski and I. Sciriha, Existence of regular nut graphs and the Fowler construction, Applicable Analysis and Discrete Mathematics, 17(2):321-333, 2023.

[3] P.W. Fowler, J.B. Gauci, J. Goedgebeur, T. Pisanski and I. Sciriha, Existence of regular nut graphs for degree at most 11, Discussiones Mathematicae Graph Theory, 40(2):533-557, 2020.


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