A184940 -id:A184940 - cp's OEIS frontend

A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.

1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, 265, 1544, 10778, 88168, 805491, 8037418, 86221634, 985870522, 11946487647, 152808063181, 2056692014474, 29051272833609, 429668180677439, 6640165204855036, 107026584471569605, 1796101588825595008, 31333997930603283531, 567437240683788292989

Offset: 0

N. J. A. Sloane

The null graph on 0 vertices is vacuously connected and 4-regular. - Jason Kimberley, Jan 29 2011
The Multiset Transform of this sequence gives a triangle which gives in row n and column k the 4-regular simple graphs with n>=1 nodes and k>=1 components (row sums A033301), starting:
;
;
;
;
1 ;
1 ;
2 ;
6 ;
16 ;
59 1 ;
265 1 ;
1544 3 ;
10778 8 ;
88168 25 ;
805491 87 1 ;
8037418 377 1 ;
86221634 2023 3 ;
985870522 13342 9 ;
11946487647 104568 27 ;
152808063181 930489 96 1 ;  - R. J. Mathar, Jun 02 2022

CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Wayne Barrett, Shaun Fallat, Veronika Furst, Shahla Nasserasr, Brendan Rooney, and Michael Tait, Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues, arXiv:2305.10562 [math.CO], 2023. See p. 7.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
M. Meringer, GenReg, Generation of regular graphs, program.
Markus Meringer, H. James Cleaves, Stephen J. Freeland, Beyond Terrestrial Biology: Charting the Chemical Universe of α-Amino Acid Structures, Journal of Chemical Information and Modeling, 53.11 (2013), pp. 2851-2862.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Quartic Graph
Eric Weisstein's World of Mathematics, Regular Graph
Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, and Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.

From Jason Kimberley, Mar 27 2010 and Jan 29 2011: (Start)
4-regular simple graphs: this sequence (connected), A033483 (disconnected), A033301 (not necessarily connected).
Connected regular simple graphs: A005177 (any degree), A068934 (triangular array); specified degree k: A002851 (k=3), this sequence (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 4-regular simple graphs with girth at least g: this sequence (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5).
Connected 4-regular graphs: this sequence (simple), A085549 (multigraphs with loops allowed), A129417  (multigraphs with loops verboten). (End)

a(n) = A184943(n) + A033886(n).
a(n) = A033301(n) - A033483(n).
Inverse Euler transform of A033301.
Row sums of A184940. - R. J. Mathar, May 30 2022

a(19)-a(22) were appended by Jason Kimberley on Sep 04 2009, Nov 24 2009, Mar 27 2010, and Mar 18 2011, from running M. Meringer's GENREG for 3.4, 44, and 403 processor days, and 15.5 processor years, at U. Ncle.

a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 10 2020

A198303 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 1, 3, 2, 13, 5, 1, 63, 20, 2, 399, 101, 8, 1, 3268, 743, 48, 1, 33496, 7350, 450, 5, 412943, 91763, 5751, 32, 5883727, 1344782, 90553, 385, 94159721, 22160335, 1612905, 7573, 1, 1661723296, 401278984, 31297357, 181224, 3, 31954666517

Offset: 2

Jason Kimberley, Nov 16 2011

The first column is for girth exactly 3. The row length is incremented to g-2 when 2n reaches A000066(g).

			1;
1, 1;
3, 2;
13, 5, 1;
63, 20, 2;
399, 101, 8, 1;
3268, 743, 48, 1;
33496, 7350, 450, 5;
412943, 91763, 5751, 32;
5883727, 1344782, 90553, 385;
94159721, 22160335, 1612905, 7573, 1;
1661723296, 401278984, 31297357, 181224, 3;
31954666517, 7885687604, 652159389, 4624480, 21;
663988090257, 166870266608, 14499780660, 122089998, 545;
14814445040728, 3781101495300, 342646718608, 3328899586, 30368;

F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A002851(n).
Connected 3-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).
Connected 3-regular 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).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: this sequence (k=3), A184940 (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).

A184941 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth at least g.

Original entry on oeis.org

1, 1, 2, 6, 1, 16, 0, 59, 2, 265, 2, 1544, 12, 10778, 31, 88168, 220, 805491, 1606, 8037418, 16828, 86221634, 193900, 985870522, 2452818, 11946487647, 32670330, 1, 152808063181, 456028474, 2, 2056692014474, 6636066099, 8, 28566273166527, 100135577747, 131

Offset: 5

Jason Kimberley, Jan 10 2012

The first column is for girth at least 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;
6, 1;
16, 0;
59, 2;
265, 2;
1544, 12;
10778, 31;
88168, 220;
805491, 1606;
8037418, 16828;
86221634, 193900;
985870522, 2452818;
11946487647, 32670330, 1;
152808063181, 456028474, 2;
2056692014474, 6636066099, 8;
28566273166527, 100135577747, 131;

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

Connected 4-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184940 (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 at least g: A185131 (k=3), this sequence (k=4), A184951 (k=5), A184961 (k=6), A184971 (k=7), A184981 (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).

cp's OEIS Frontend

A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.

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A198303 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n 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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A184941 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth at least 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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