A006926
Number of connected trivalent graphs with 2n nodes and girth exactly 6.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 32, 385, 7573, 181224, 4624480, 122089998, 3328899586, 93988909755
Offset: 0
- CRC Handbook of Combinatorial Designs, 1996, p. 647.
- Gordon Royle, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Definition corrected to include "connected", and "girth at least 6" minus "girth at least 7" formula provided by
Jason Kimberley, Dec 12 2009
A006927
Number of connected trivalent graphs with 2n nodes and girth exactly 7.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 21, 545, 30368, 1782839, 95079080, 4686063107
Offset: 0
- Gordon Royle, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Definition amended to include "connected" (no disconnected yet), and "girth at least 7" minus "girth at least 8" formula provided by
Jason Kimberley, Dec 12 2009
A184945
Number of connected 4-regular simple graphs on n vertices with girth exactly 5.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 8, 131, 3917, 123859, 4131991, 132160607, 4018022149, 118369811959
Offset: 0
a(0)=0 because even though the null graph (on zero vertices) is vacuously 4-regular and connected, since it is acyclic, it has infinite girth.
The a(19)=1 graph is the unique (4,5) cage: the Robertson graph (see also A159191). It has the following adjacency lists.
01 : 02 03 04 05
02 : 01 06 07 08
03 : 01 09 10 11
04 : 01 12 13 14
05 : 01 15 16 17
06 : 02 09 12 15
07 : 02 10 13 16
08 : 02 11 14 17
09 : 03 06 13 17
10 : 03 07 14 18
11 : 03 08 16 19
12 : 04 06 16 18
13 : 04 07 09 19
14 : 04 08 10 15
15 : 05 06 14 19
16 : 05 07 11 12
17 : 05 08 09 18
18 : 10 12 17 19
19 : 11 13 15 18
4-regular simple graphs with girth exactly 5: this sequence (connected),
A185045 (disconnected),
A185145 (not necessarily connected).
Connected k-regular simple graphs with girth exactly 5:
A006925 (k=3), this sequence (k=4),
A184955 (k=5).
Connected 4-regular simple graphs with girth exactly g:
A184943 (g=3),
A184944 (g=4), this sequence (g=5).
A184955
Number of connected 5-regular simple graphs on 2n vertices with girth exactly 5.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 90
Offset: 0
Connected k-regular simple graphs with girth exactly 5:
A185015 (k=2),
A006925 (k=3),
A184945 (k=4), this sequence (k=5).
Connected 5-regular simple graphs with girth exactly g:
A184953 (g=3),
A184954 (g=4), this sequence (g=5).
A185135
Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 5.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 2, 8, 48, 450, 5752, 90555, 1612917, 31297424, 652159986, 14499787794, 342646826428
Offset: 0
Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g:
A185130 (triangle); fixed g:
A185133 (g=3),
A185134 (g=4), this sequence (g=5),
A185136 (g=6).
A210709
Number of trivalent connected simple graphs with 2n nodes and girth at least 9.
Original entry on oeis.org
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18
Offset: 0