House of Graphs

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:

Fullerenes
Fullerenes without a spiral starting at a pentagon
Fullerenes without a spiral

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
20	12	1	0	0	0	0
22	13	0	0	0	0	0
24	14	1	0	0	0	0
26	15	1	0	0	0	0
28	16	2	0	0	0	0
30	17	3	0	0	0	0
32	18	6	0	0	0	0
34	19	6	0	0	0	0
36	20	15	0	0	0	0
38	21	17	0	0	0	0
40	22	40	0	0	0	0
42	23	45	0	0	0	0
44	24	89	0	0	0	0
46	25	116	0	0	0	0
48	26	199	0	0	0	0
50	27	271	0	0	0	0
52	28	437	0	0	0	0
54	29	580	0	0	0	0
56	30	924	0	0	0	0
58	31	1205	0	0	0	0
60	32	1812	1	0	0	0
62	33	2385	0	0	0	0
64	34	3465	0	0	0	0
66	35	4478	0	0	0	0
68	36	6332	0	0	0	0
70	37	8149	1	0	0	0
72	38	11190	1	0	0	0
74	39	14246	1	0	0	0
76	40	19151	2	0	0	0
78	41	24109	5	0	0	0
80	42	31924	7	0	0	0
82	43	39718	9	0	0	0
84	44	51592	24	0	0	0
86	45	63761	19	0	0	0
88	46	81738	35	0	0	0
90	47	99918	46	0	0	0
92	48	126409	86	0	0	0
94	49	153493	134	0	0	0
96	50	191839	187	0	0	0
98	51	231017	259	0	0	0
100	52	285914	450	0	0	0
102	53	341658	616	0	0	0
104	54	419013	823	0	0	0
106	55	497529	1233	0	0	0
108	56	604217	1799	0	0	0
110	57	713319	2355	0	0	0
112	58	860161	3342	0	0	0
114	59	1008444	4468	0	0	0
116	60	1207119	6063	0	0	0
118	61	1408553	8148	0	0	0
120	62	1674171	10774	0	0	0
122	63	1942929	13977	0	0	0
124	64	2295721	18769	0	0	0
126	65	2650866	23589	0	0	0
128	66	3114236	30683	0	0	0
130	67	3580637	39393	0	0	0
132	68	4182071	49878	0	0	0
134	69	4787715	62372	0	0	0
136	70	5566949	79362	0	0	0
138	71	6344698	98541	0	0	0
140	72	7341204	121354	1	0	0
142	73	8339033	151201	0	0	0
144	74	9604411	186611	0	0	0
146	75	10867631	225245	0	0	0
148	76	12469092	277930	0	0	0
150	77	14059174	335569	1	0	0
152	78	16066025	404667	2	0	0
154	79	18060979	489646	0	0	0
156	80	20558767	586264	0	0	0
158	81	23037594	697720	0	0	0
160	82	26142839	836497	2	0	0
162	83	29202543	989495	1	0	0
164	84	33022573	1170157	2	0	0
166	85	36798433	1382953	1	0	0
168	86	41478344	1628029	13	0	0
170	87	46088157	1902265	4	0	0
172	88	51809031	2234133	12	0	0
174	89	57417264	2601868	10	0	0
176	90	64353269	3024383	28	0	0
178	91	71163452	3516365	23	0	0
180	92	79538751	4071832	58	0	0
182	93	87738311	4690880	54	0	0
184	94	97841183	5424777	142	0	0
186	95	107679717	6229550	129	0	0
188	96	119761075	7144091	291	0	0
190	97	131561744	8187581	257	0	0
192	98	145976674	9364975	548	0	0
194	99	159999462	10659863	566	0	0
196	100	177175687	12163298	1126	0	0
198	101	193814658	13809901	1072	0	0
200	102	214127742	15655672	1943	0	0
202	103	233846463	17749388	2080	0	0
204	104	257815889	20070486	3682	0	0
206	105	281006325	22606939	3992	0	0
208	106	309273526	25536557	6340	0	0
210	107	336500830	28700677	6737	0	0
212	108	369580714	32230861	10513	0	0
214	109	401535955	36173081	12000	0	0
216	110	440216206	40536922	18169	0	0
218	111	477420176	45278722	20019	0	0
220	112	522599564	50651799	28528	0	0
222	113	565900181	56463948	32276	0	0
224	114	618309598	62887775	46534	0	0
226	115	668662698	69995887	52177	0	0
228	116	729414880	77831323	71303	0	0
230	117	787556069	86238206	79915	0	0
232	118	857934016	95758929	109848	0	0
234	119	925042498	105965373	124153	0	0
236	120	1006016526	117166528	164700	0	0
238	121	1083451816	129476607	184404	0	0
240	122	1176632247	142960479	242507	1	0
242	123	1265323971	157402781	273885	0	0
244	124	1372440782	173577766	353997	0	0
246	125	1474111053	190809628	397673	0	0
248	126	1596482232	209715141	507913	0	0
250	127	1712934069	230272559	570053	0	0
252	128	1852762875	252745513	717983	0	0
254	129	1985250572	276599787	805374	0	0
256	130	2144943655	303235792	1007680	0	0
258	131	2295793276	331516984	1127989	0	0
260	132	2477017558	362302637	1392996	2	0
262	133	2648697036	395600325	1550580	0	0
264	134	2854536850	431894257	1905849	0	0
266	135	3048609900	470256444	2124873	1	0
268	136	3282202941	512858451	2592104	1	0
270	137	3501931260	557745670	2868467	2	0
272	138	3765465341	606668511	3461487	1	0
274	139	4014007928	659140287	3847594	0	0
276	140	4311652376	716217922	4621524	1	0
278	141	4591045471	776165188	5112067	2	0
280	142	4926987377	842498881	6079570	4	0
282	143	5241548270	912274540	6726996	1	0
284	144	5618445787	987874095	7971111	10	0
286	145	5972426835	1068507788	8784514	3	0
288	146	6395981131	1156161307	10352546	7	0
290	147	6791769082	1247686189	11385724	9	0
292	148	7267283603	1348832364	13357318	5	0
294	149	7710782991	1454359806	14652198	6	0
296	150	8241719706	1568768524	17102231	24	0
298	151	8738236515	1690214836	18756139	16	0
300	152	9332065811	1821766896	21766152	32	0
302	153	9884604767	1958581588	23815310	36	0
304	154	10548218751	2109271290	27529516	46	0
306	155	11164542762	2266138871	30090574	54	0
308	156	11902015724	2435848971	34629672	99	0
310	157	12588998862	2614544391	37770691	93	0
312	158	13410330482	2808510141	43312313	135	0
314	159	14171344797	3009120113	47153778	187	0
316	160	15085164571	3229731630	53899686	211	0
318	161	15930619304	3458148016	58585441	308	0
320	162	16942010457	3704939275	66712070	443	0
322	163	17880232383	3964153268	72395888	535	0
324	164	19002055537	4244706701	82171212	698	0
326	165	20037346408	4533465777	89063353	1026	0
328	166	21280571390	4850870260	100785130	1216	0
330	167	22426253115	5178120469	109068073	1623	0
332	168	23796620378	5531727283	122992213	2489	0
334	169	25063227406	5900369830	132950223	2788	0
336	170	26577912084	6299880577	149523121	3612	0
338	171	27970034826	6709574675	161430830	4744	0
340	172	29642262229	7158963073	181076418	5845	0
342	173	31177474996	7620446934	195124334	7457	0
344	174	33014225318	8118481242	218323289	10591	0
346	175	34705254287	8636262789	235050400	12307	0
348	176	36728266430	9196920285	262381050	15312	0
350	177	38580626759	9768511147	282042413	19574	0
352	178	40806395661	10396040696	314052518	23755	0
354	179	42842199753	11037658075	337229970	29793	0
356	180	45278616586	11730538496	374666300	38688	0
358	181	47513679057	12446446419	401932458	45946	0
360	182	50189039868	13221751502	445482235	55742	0
362	183	52628839448	14010515381	477264068	69970	0
364	184	55562506886	14874753568	528016753	83616	0
366	185	58236270451	15754940959	565045586	100644	0
368	186	61437700788	16705334454	623895236	126048	0
370	187	64363670678	17683643273	666935811	149044	0
372	188	67868149215	18744292915	734907336	179013	0
374	189	71052718441	19816289281	784797263	217673	0
376	190	74884539987	20992425825	863237405	257673	0
378	191	78364039771	22186413139	920935351	302553	0
380	192	82532990559	23475079272	1011152383	367547	1
382	193	86329680991	24795898388	1077679749	434339	0
384	194	90881152117	26227197453	1181149036	507481	0
386	195	95001297565	27670862550	1257630423	611532	0
388	196	99963147805	29254036711	1376400812	707184	0
390	197	104453597992	30852950986	1463926563	820525	0
392	198	109837310021	32581366295	1599524989	982532	0
394	199	114722988623	34345173894	1699970613	1133377	0
396	200	120585261143	36259212641	1854374011	1323509	0
398	201	125873325588	38179777473	1969147856	1546304	0
400	202	132247999328	40286153024	2144985583	1784313	1

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Fullerenes without spiral starting at pentagon

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

Vertices	Faces	No pentagon spiral
100	52	1
102	53	0
104	54	1
106	55	0
108	56	0
110	57	1
112	58	0
114	59	0
116	60	0
118	61	0
120	62	0
122	63	0
124	64	1
126	65	0
128	66	0
130	67	0
132	68	1
134	69	1
136	70	1
138	71	0
140	72	1
142	73	0
144	74	2
146	75	0
148	76	2
150	77	0
152	78	3
154	79	0
156	80	3
158	81	0
160	82	1
162	83	1
164	84	10
166	85	3
168	86	10
170	87	4
172	88	8
174	89	2
176	90	14
178	91	6
180	92	8
182	93	9
184	94	16
186	95	10
188	96	18
190	97	6
192	98	33
194	99	13
196	100	34
198	101	18
200	102	36
202	103	25
204	104	59
206	105	35
208	106	66
210	107	37
212	108	89
214	109	57
216	110	85
218	111	62
220	112	87
222	113	64
224	114	172
226	115	99
228	116	198
230	117	141
232	118	194
234	119	141
236	120	316
238	121	205
240	122	400
242	123	259
244	124	468
246	125	397
248	126	634
250	127	411
252	128	615
254	129	467
256	130	851
258	131	562
260	132	881
262	133	623
264	134	1083
266	135	863
268	136	1270
270	137	1037
272	138	1558
274	139	1133
276	140	1968
278	141	1525
280	142	2529
282	143	2002
284	144	3011
286	145	2473
288	146	3413
290	147	2783
292	148	4215
294	149	3401
296	150	4996
298	151	3797
300	152	5548
302	153	4388
304	154	6193
306	155	4938
308	156	7673
310	157	6242
312	158	9165
314	159	7261
316	160	10302
318	161	8464
320	162	11854
322	163	9745
324	164	14356
326	165	12344
328	166	17926
330	167	15397
332	168	21182
334	169	17986
336	170	23625
338	171	19571
340	172	26885
342	173	22801
344	174	31476
346	175	26842
348	176	35834
350	177	30885
352	178	41747
354	179	35180
356	180	47021
358	181	39661
360	182	51978
362	183	44499
364	184	57767
366	185	50370
368	186	66261
370	187	58003
372	188	77534
374	189	68670
376	190	89284
378	191	77802
380	192	100355
382	193	86960
384	194	112914
386	195	101046
388	196	131212
390	197	117963
392	198	152483
394	199	134408
396	200	171302
398	201	150285
400	202	189662

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Fullerenes without a spiral

All fullerenes with less than 380 vertices have a spiral.

Vertices	Faces	No spiral
380	192	1
382	193	0
384	194	1
386	195	0
388	196	0
390	197	0
392	198	0
394	199	0
396	200	0
398	201	0
400	202	0

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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.

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Vertices	Faces	No spiral
380	192	1
382	193	0
384	194	1
386	195	0
388	196	0
390	197	0
392	198	0
394	199	0
396	200	0
398	201	0
400	202	0

Vertices	Faces	No spiral
380	192	1
382	193	0
384	194	1
386	195	0
388	196	0
390	197	0
392	198	0
394	199	0
396	200	0
398	201	0
400	202	0

Vertices	Faces	No spiral
380	192	1
382	193	0
384	194	1
386	195	0
388	196	0
390	197	0
392	198	0
394	199	0
396	200	0
398	201	0
400	202	0