A198303 -id:A198303 - cp's OEIS frontend

A006926 Number of connected trivalent graphs with 2n nodes and girth exactly 6.

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).

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); specified g: A006923 (g=3), A006924 (g=4), A006925 (g=5), this sequence (g=6), A006927 (g=7).

Connected 3-regular simple graphs with girth at least g: A185131 (triangle); A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

a(n) = A014374(n) - A014375(n).

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

N. J. A. Sloane

Gordon Royle, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); specified g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), this sequence (g=7).

Connected 3-regular simple graphs with girth at least g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

a(n) = A014375(n) - A014376(n).

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

A184940 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 2, 5, 1, 16, 0, 57, 2, 263, 2, 1532, 12, 10747, 31, 87948, 220, 803885, 1606, 8020590, 16828, 86027734, 193900, 983417704, 2452818, 11913817317, 32670329, 1, 152352034707, 456028472, 2, 2050055948375, 6636066091, 8, 28466137588780, 100135577616, 131

Offset: 5

Jason Kimberley, Feb 24 2011

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4. The row length is incremented to g-2 when n reaches A037233(g).

			1;
1;
2;
5, 1;
16, 0;
57, 2;
263, 2;
1532, 12;
10747, 31;
87948, 220;
803885, 1606;
8020590, 16828;
86027734, 193900;
983417704, 2452818;
11913817317, 32670329, 1;
152352034707, 456028472, 2;
2050055948375, 6636066091, 8;
28466137588780, 100135577616, 131;

Jason Kimberley, Incomplete table of i, n, g, C(n,g)=a(i) for row n = 5..36
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Connected 4-regular simple graphs with girth at least g: A184941 (triangle); chosen g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184943 (g=3), A184944 (g=4), A184945 (g=5), A184946 (g=6).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), this sequence (k=4), A184950 (k=5), A184960 (k=6), A184970 (k=7), A184980 (k=8).

A184980 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105934, 1

Offset: 9

Jason Kimberley, Jan 19 2012

The first column is for girth at least 3. The row length is incremented to g-2 when 2n reaches A054760(8,g).

			1;
1;
6;
94;
10786;
3459386;
1470293676;
733351105934, 1;
?, 0;
?, 1;
?, 0;
?, 13;
?, 1;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184983 (g=3).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), A184960 (k=6), A184970 (k=7), this sequence (k=8).

A184950 Irregular triangle C(n,g) counting the connected 5-regular simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 3, 59, 1, 7847, 1, 3459376, 7, 2585136287, 388, 2807104844073, 406824

Offset: 3

Jason Kimberley, Feb 24 2011

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4. The row length is incremented to g-2 when 2n reaches A054760(5,g).

			1;
3;
59, 1;
7847, 1;
3459376, 7;
2585136287, 388;
2807104844073, 406824;
?, 1125022325;
?, 3813549359274;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g
Jason Kimberley, Incomplete table of i, n, g, C(n,g)=a(i) for row n = 3..22

Connected 5-regular simple graphs with girth at least g: A184951 (triangle); chosen g: A006821 (g=3), A058275 (g=4).
Connected 5-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184953 (g=3), A184954 (g=4), A184955 (g=5).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), this sequence (k=5), A184960 (k=6), A184970 (k=7), A184980 (k=8).

A184960 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 4, 21, 266, 7848, 1, 367860, 0, 21609299, 1, 1470293674, 1, 113314233799, 9, 9799685588930, 6

Offset: 7

Jason Kimberley, Feb 24 2011

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3. The row length is incremented to g-2 when n reaches A054760(6,g).

			Triangle begins:
1;
1;
4;
21;
266;
7848, 1;
367860, 0;
21609299, 1;
1470293674, 1;
113314233799, 9;
9799685588930, 6;
?, 267;
?, 3727;
?, 483012;
?, 69823723;
?, 14836130862;
The C(40,5)=1 (see the a-file) graph, the unique (6,5)-cage, is the complement of a Petersen graph inside the Hoffman-Singleton graph [from Brouwer link].
The first known of C(42,5)>=1 graph(s) has automorphism group of order 5040 and these adjacency lists:
1 : 2 3 4 5 6 7
2 : 1 8 9 10 11 12
3 : 1 13 14 15 16 17
4 : 1 18 19 20 21 22
5 : 1 23 24 25 26 27
6 : 1 28 29 30 31 32
7 : 1 33 34 35 36 37
8 : 2 13 18 23 28 38
9 : 2 14 19 24 33 39
10 : 2 15 20 29 34 40
11 : 2 16 25 30 35 41
12 : 2 21 26 31 36 42
13 : 3 8 21 27 34 41
14 : 3 9 26 28 37 40
15 : 3 10 22 25 31 39
16 : 3 11 19 32 36 38
17 : 3 20 23 30 33 42
18 : 4 8 25 32 33 40
19 : 4 9 16 27 29 42
20 : 4 10 17 26 35 38
21 : 4 12 13 30 37 39
22 : 4 15 24 28 36 41
23 : 5 8 17 29 36 39
24 : 5 9 22 30 34 38
25 : 5 11 15 18 37 42
26 : 5 12 14 20 32 41
27 : 5 13 19 31 35 40
28 : 6 8 14 22 35 42
29 : 6 10 19 23 37 41
30 : 6 11 17 21 24 40
31 : 6 12 15 27 33 38
32 : 6 16 18 26 34 39
33 : 7 9 17 18 31 41
34 : 7 10 13 24 32 42
35 : 7 11 20 27 28 39
36 : 7 12 16 22 23 40
37 : 7 14 21 25 29 38
38 : 8 16 20 24 31 37
39 : 9 15 21 23 32 35
40 : 10 14 18 27 30 36
41 : 11 13 22 26 29 33
42 : 12 17 19 25 28 34

Andries E. Brouwer, Cages
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g
Jason Kimberley, Incomplete table of i, n, g, C(n,g)=a(i) for row n = 7..42

Connected 6-regular simple graphs with girth at least g: A184961 (triangle); chosen g: A006822 (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184963 (g=3), A184964 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), this sequence (k=6), A184970 (k=7), A184980 (k=8).

After approximately 390 processor days of computation with genreg, C(41,5)=0.

A184970 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 5, 1547, 21609300, 1, 733351105933, 1

Offset: 4

Jason Kimberley, Feb 25 2011

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2. The row length is incremented to g-2 when 2n reaches A054760(7,g).

			1;
5;
1547;
21609300, 1;
733351105933, 1;
?, 8;
?, 741;
?, 2887493;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g
Jason Kimberley, Incomplete table of i, n, g, C(n,g)=a(i) for row n = 4..11

Connected 7-regular simple graphs with girth at least g: A184971 (triangle); chosen g: A014377 (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184973 (g=3), A184974 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), A184960 (k=6), this sequence (k=7), A184980 (k=8).

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

Jason Kimberley, Dec 20 2012

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g

Trivalent simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8), this sequence (g=9).

Trivalent simple graphs with girth exactly g: A198303 (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).

a(29) = a(A000066(9)/2) = A052453(9) = 18 is the number of (3,9) cages.

cp's OEIS Frontend

A006926 Number of connected trivalent graphs with 2n nodes and girth exactly 6.

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A006927 Number of connected trivalent graphs with 2n nodes and girth exactly 7.

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A184940 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth exactly g.

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A184980 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth exactly g.

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A184950 Irregular triangle C(n,g) counting the connected 5-regular simple graphs on 2n vertices with girth exactly g.

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A184960 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth exactly g.

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A184970 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth exactly g.

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A210709 Number of trivalent connected simple graphs with 2n nodes and girth at least 9.

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