# Fullerenes

Fullerenes are cubic plane graphs where all faces are pentagons or hexagons. Euler's formula implies that each fullerene contains exactly 12 pentagonal faces. The dual graph of a fullerene is a triangulation where all vertices have degree 5 or 6.

The fullerene lists are currently only available in 'planar_code' format. The larger files are compressed with gzip.

The following lists are available:

The 'face-distance' between two pentagons is the distance between the corresponding vertices of degree 5 in the dual graph. We refer to the least face-distance between pentagons of a fullerene as the 'pentagon separation' of the fullerene. The table below lists the number of fullerenes up to 400 vertices with pentagon separation at least d. Note that d=1 gives the set of all fullerenes and d=2 gives the set of all IPR fullerenes. More information about the pentagon separation of fullerenes can be found in [1].

All fullerene and IPR fullerene counts up to 380 vertices were independently confirmed by fullgen and buckygen. The fullerenes with more than 380 vertices were generated with buckygen.

The fullerenes in the downloadable lists from the "Fullerenes"-table are sorted according to their lexicographically minimal spiral development. So the order in which they appear is the same as in the Atlas of Fullerenes [2]. More information about spirals in fullerenes can be found in [3].

### Fullerenes

Vertices Faces Fullerenes IPR Fullerenes Pent. sep. ≥ 3 Pent. sep. ≥ 4 Pent. sep. ≥ 5
201210000
221300000
241410000
261510000
281620000
301730000
321860000
341960000
3620150000
3821170000
4022400000
4223450000
4424890000
46251160000
48261990000
50272710000
52284370000
54295800000
56309240000
583112050000
603218121000
623323850000
643434650000
663544780000
683663320000
703781491000
7238111901000
7439142461000
7640191512000
7841241095000
8042319247000
8243397189000
84445159224000
86456376119000
88468173835000
90479991846000
924812640986000
9449153493134000
9650191839187000
9851231017259000
10052285914450000
10253341658616000
10454419013823000
106554975291233000
108566042171799000
110577133192355000
112588601613342000
1145910084444468000
1166012071196063000
1186114085538148000
12062167417110774000
12263194292913977000
12464229572118769000
12665265086623589000
12866311423630683000
13067358063739393000
13268418207149878000
13469478771562372000
13670556694979362000
13871634469898541000
140727341204121354100
142738339033151201000
144749604411186611000
1467510867631225245000
1487612469092277930000
1507714059174335569100
1527816066025404667200
1547918060979489646000
1568020558767586264000
1588123037594697720000
1608226142839836497200
1628329202543989495100
16484330225731170157200
16685367984331382953100
168864147834416280291300
17087460881571902265400
172885180903122341331200
174895741726426018681000
176906435326930243832800
178917116345235163652300
180927953875140718325800
182938773831146908805400
1849497841183542477714200
18695107679717622955012900
18896119761075714409129100
19097131561744818758125700
19298145976674936497554800
194991599994621065986356600
19610017717568712163298112600
19810119381465813809901107200
20010221412774215655672194300
20210323384646317749388208000
20410425781588920070486368200
20610528100632522606939399200
20810630927352625536557634000
21010733650083028700677673700
212108369580714322308611051300
214109401535955361730811200000
216110440216206405369221816900
218111477420176452787222001900
220112522599564506517992852800
222113565900181564639483227600
224114618309598628877754653400
226115668662698699958875217700
228116729414880778313237130300
230117787556069862382067991500
2321188579340169575892910984800
23411992504249810596537312415300
236120100601652611716652816470000
238121108345181612947660718440400
240122117663224714296047924250710
242123126532397115740278127388500
244124137244078217357776635399700
246125147411105319080962839767300
248126159648223220971514150791300
250127171293406923027255957005300
252128185276287525274551371798300
254129198525057227659978780537400
2561302144943655303235792100768000
2581312295793276331516984112798900
2601322477017558362302637139299620
2621332648697036395600325155058000
2641342854536850431894257190584900
2661353048609900470256444212487310
2681363282202941512858451259210410
2701373501931260557745670286846720
2721383765465341606668511346148710
2741394014007928659140287384759400
2761404311652376716217922462152410
2781414591045471776165188511206720
2801424926987377842498881607957040
2821435241548270912274540672699610
28414456184457879878740957971111100
28614559724268351068507788878451430
288146639598113111561613071035254670
290147679176908212476861891138572490
292148726728360313488323641335731850
294149771078299114543598061465219860
2961508241719706156876852417102231240
2981518738236515169021483618756139160
3001529332065811182176689621766152320
3021539884604767195858158823815310360
30415410548218751210927129027529516460
30615511164542762226613887130090574540
30815611902015724243584897134629672990
31015712588998862261454439137770691930
312158134103304822808510141433123131350
314159141713447973009120113471537781870
316160150851645713229731630538996862110
318161159306193043458148016585854413080
320162169420104573704939275667120704430
322163178802323833964153268723958885350
324164190020555374244706701821712126980
3261652003734640845334657778906335310260
32816621280571390485087026010078513012160
33016722426253115517812046910906807316230
33216823796620378553172728312299221324890
33416925063227406590036983013295022327880
33617026577912084629988057714952312136120
33817127970034826670957467516143083047440
34017229642262229715896307318107641858450
34217331177474996762044693419512433474570
344174330142253188118481242218323289105910
346175347052542878636262789235050400123070
348176367282664309196920285262381050153120
350177385806267599768511147282042413195740
3521784080639566110396040696314052518237550
3541794284219975311037658075337229970297930
3561804527861658611730538496374666300386880
3581814751367905712446446419401932458459460
3601825018903986813221751502445482235557420
3621835262883944814010515381477264068699700
3641845556250688614874753568528016753836160
36618558236270451157549409595650455861006440
36818661437700788167053344546238952361260480
37018764363670678176836432736669358111490440
37218867868149215187442929157349073361790130
37418971052718441198162892817847972632176730
37619074884539987209924258258632374052576730
37819178364039771221864131399209353513025530
380192825329905592347507927210111523833675471
382193863296809912479589838810776797494343390
384194908811521172622719745311811490365074810
386195950012975652767086255012576304236115320
388196999631478052925403671113764008127071840
3901971044535979923085295098614639265638205250
3921981098373100213258136629515995249899825320
39419911472298862334345173894169997061311333770
39620012058526114336259212641185437401113235090
39820112587332558838179777473196914785615463040
40020213224799932840286153024214498558317843131

### Fullerenes without a spiral starting at a pentagon

All fullerenes with less than 100 vertices have a spiral starting at a pentagon.

Vertices Faces No pentagon
spiral
100521
102530
104541
106550
108560
110571
112580
114590
116600
118610
120620
122630
124641
126650
128660
130670
132681
134691
136701
138710
140721
142730
144742
146750
148762
150770
152783
154790
156803
158810
160821
162831
1648410
166853
1688610
170874
172888
174892
1769014
178916
180928
182939
1849416
1869510
1889618
190976
1929833
1949913
19610034
19810118
20010236
20210325
20410459
20610535
20810666
21010737
21210889
21410957
21611085
21811162
22011287
22211364
224114172
22611599
228116198
230117141
232118194
234119141
236120316
238121205
240122400
242123259
244124468
246125397
248126634
250127411
252128615
254129467
256130851
258131562
260132881
262133623
2641341083
266135863
2681361270
2701371037
2721381558
2741391133
2761401968
2781411525
2801422529
2821432002
2841443011
2861452473
2881463413
2901472783
2921484215
2941493401
2961504996
2981513797
3001525548
3021534388
3041546193
3061554938
3081567673
3101576242
3121589165
3141597261
31616010302
3181618464
32016211854
3221639745
32416414356
32616512344
32816617926
33016715397
33216821182
33416917986
33617023625
33817119571
34017226885
34217322801
34417431476
34617526842
34817635834
35017730885
35217841747
35417935180
35618047021
35818139661
36018251978
36218344499
36418457767
36618550370
36818666261
37018758003
37218877534
37418968670
37619089284
37819177802
380192100355
38219386960
384194112914
386195101046
388196131212
390197117963
392198152483
394199134408
396200171302
398201150285
400202189662

### Fullerenes without a spiral

All fullerenes with less than 380 vertices have a spiral.

Vertices Faces No spiral
3801921
3821930
3841941
3861950
3881960
3901970
3921980
3941990
3962000
3982010
4002020

## References

[1] J. Goedgebeur and B.D. McKay, Fullerenes with distant pentagons, MATCH Commun. Math. Comput. Chem., 74(3):659-672, 2015.
[2] P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Clarendon press, Oxford, 1995.
[3] G. Brinkmann, J. Goedgebeur and B.D. McKay, The smallest fullerene without a spiral, Chemical Physics Letters, 522:54-55, 2012.