A002851 -id:A002851 - 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

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 6, 21, 94, 540, 4207, 42110, 516344, 7373924, 118573592, 2103205738, 40634185402, 847871397424, 18987149095005, 454032821688754, 11544329612485981, 310964453836198311, 8845303172513781271

Offset: 0

N. J. A. Sloane

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-4)-regular graphs on 2n vertices.

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).

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Jason Kimberley, Not-necessarily connected regular graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
R. W. Robinson, Cubic graphs (notes)
Robinson, R. W.; Wormald, N. C., Numbers of cubic graphs, J. Graph Theory 7 (1983), no. 4, 463-467.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Cubic Graph
Gal Weitz, Lirandë Pira, Chris Ferrie, and Joshua Combes, Sub-universal variational circuits for combinatorial optimization problems, arXiv:2308.14981 [quant-ph], 2023.

Cf. A000421.
Row sums of A275744.
3-regular simple graphs: A002851 (connected), A165653 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), this sequence (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Not necessarily connected 3-regular simple graphs with girth *at least* g: this sequence (g=3), A185334 (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A002851(n) + A165653(n).

This sequence is the Euler transformation of A002851.

More terms from Ronald C. Read.

Comment, formulas, and (most) crossrefs by Jason Kimberley, 2009 and 2012

A068934 Triangular array C(n, r) = number of connected r-regular graphs with n nodes, 0 <= r < n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 5, 6, 3, 1, 1, 0, 0, 1, 0, 16, 0, 4, 0, 1, 0, 0, 1, 19, 59, 60, 21, 5, 1, 1, 0, 0, 1, 0, 265, 0, 266, 0, 6, 0, 1, 0, 0, 1, 85, 1544, 7848, 7849, 1547, 94, 9, 1, 1, 0, 0, 1, 0, 10778, 0, 367860, 0

Offset: 1

David Wasserman, Mar 08 2002

A graph is called r-regular if every node has exactly r edges. The numbers in this table were copied from the column sequences.

This sequence can be derived from A051031 by inverse Euler transform. See the comments in A051031 for a brief description of how that sequence can be computed without generating all regular graphs. - Andrew Howroyd, Mar 13 2020

			01: 1;
02: 0, 1;
03: 0, 0, 1;
04: 0, 0, 1, 1;
05: 0, 0, 1, 0, 1;
06: 0, 0, 1, 2, 1, 1;
07: 0, 0, 1, 0, 2, 0, 1;
08: 0, 0, 1, 5, 6, 3, 1, 1;
09: 0, 0, 1, 0, 16, 0, 4, 0, 1;
10: 0, 0, 1, 19, 59, 60, 21, 5, 1, 1;
11: 0, 0, 1, 0, 265, 0, 266, 0, 6, 0, 1;
12: 0, 0, 1, 85, 1544, 7848, 7849, 1547, 94, 9, 1, 1;
13: 0, 0, 1, 0, 10778, 0, 367860, 0, 10786, 0, 10, 0, 1;
14: 0, 0, 1, 509, 88168, 3459383, 21609300, 21609301, 3459386, 88193, 540, 13, 1, 1;
15: 0, 0, 1, 0, 805491, 0, 1470293675, 0, 1470293676, 0, 805579, 0, 17, 0, 1;
16: 0, 0, 1, 4060, 8037418, 2585136675, 113314233808, 733351105934, 733351105935, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24, first 16 rows from Jason Kimberley)
Jason Kimberley, Connected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.

Connected regular simple graphs: A005177 (any degree -- sum of rows), this sequence (triangular array), specified degree r (columns): A002851 (r=3), A006820 (r=4), A006821 (r=5), A006822 (r=6), A014377 (r=7), A014378 (r=8), A014381 (r=9), A014382 (r=10), A014384 (r=11).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *at least* g: this sequence (g=3), A186714 (g=4), A186715 (g=5), A186716 (g=6), A186717 (g=7), A186718 (g=8), A186719 (g=9).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: A186733 (g=3), A186734 (g=4).

C(n, r) = A051031(n, r) - A068933(n, r).

Column k is the inverse Euler transform of column k of A051031. - Andrew Howroyd, Mar 10 2020

Edited by Jason Kimberley, Sep 23 2009, Nov 2011, Jan 2012, and Mar 2012

A005177 Number of connected regular graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 4, 17, 22, 167, 539, 18979, 389436, 50314796, 2942198440, 1698517036411, 442786966115560, 649978211591600286, 429712868499646474880, 2886054228478618211088773, 8835589045148342277771518309, 152929279364927228928021274993215, 1207932509391069805495173301992815105, 99162609848561525198669168640159162918815

Offset: 0

N. J. A. Sloane

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

E. Friedman, Illustration of small graphs
Daniel R. Herber, Enhancements to the perfect matching approach for graph enumeration-based engineering challenges, Proceedings of the ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE 2020).
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
Markus Meringer, GENREG: A program for Connected Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Sep 23 2009]
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Regular Graph.

Regular simple graphs of any degree: this sequence (connected), A068932 (disconnected), A005176 (not necessarily connected), A275420 (multisets).
Connected regular graphs of any degree with girth at least g: this sequence (g=3), A186724 (g=4), A186725 (g=5), A186726 (g=6), A186727 (g=7), A186728 (g=8), A186729 (g=9).
Connected regular simple graphs: this sequence (any degree), A068934 (triangular array); specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11). - Jason Kimberley, Nov 03 2011

a(n) = sum of the n-th row of A068934.
a(n) = A165647(n) - A165648(n).
This sequence is the inverse Euler transformation of A165647.

More terms from David Wasserman, Mar 08 2002
a(15) from Giovanni Resta, Feb 05 2009
Terms are sums of the output from M. Meringer's genreg software. To complete a(16) it was run by Jason Kimberley, Sep 23 2009
a(0)=1 (due to the empty graph being vacuously connected and regular) inserted by Jason Kimberley, Apr 11 2012
a(17)-a(21) from Andrew Howroyd, Mar 10 2020
a(22)-a(24) from Andrew Howroyd, May 19 2020

A014371 Number of trivalent connected simple graphs with 2*n nodes and girth at least 4.

Original entry on oeis.org

1, 0, 0, 1, 2, 6, 22, 110, 792, 7805, 97546, 1435720, 23780814, 432757568, 8542471494, 181492137812, 4127077143862

Offset: 0

N. J. A. Sloane

The null graph on 0 vertices is vacuously connected and 3-regular; since it is acyclic, it has infinite girth. - Jason Kimberley, Jan 29 2011

CRC Handbook of Combinatorial Designs, 1996, p. 647.

G. Brinkmann, J. Goedgebeur and B. D. McKay, Generation of Cubic graphs, Discrete Mathematics and Theoretical Computer Science, 13 (2) (2011), 69-80. (hal-00990486)
House of Graphs, Cubic graphs.
Jason Kimberley, Connected regular graphs with girth at least 4
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.

From Jason Kimberley, Jun 28 2010 and Jan 29 2011: (Start)
3-regular simple graphs with girth at least 4: this sequence (connected), A185234 (disconnected), A185334 (not necessarily connected).
Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), this sequence (k=3), A033886 (k=4), A058275 (k=5), A058276 (k=6), A181153 (k=7), A181154 (k=8), A181170 (k=9).
Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), this sequence (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).
Connected 3-regular 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). (End)

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A002851 = A@002851;
A006923 = A@006923;
a[n_] := A002851[[n + 1]] - A006923[[n + 1]];
a /@ Range[0, 16] (* Jean-François Alcover, Jan 27 2020 *)

Terms a(14) and a(15) appended, from running Meringer's GENREG for 4.2 and 93.2 processor days at U. Newcastle, by Jason Kimberley on Jun 28 2010

a(16), from House of Graphs, by Jan Goedgebeur et al., added by Jason Kimberley, Feb 15 2011

A006821 Number of connected regular graphs of degree 5 (or quintic graphs) with 2n nodes.

Original entry on oeis.org

1, 0, 0, 1, 3, 60, 7848, 3459383, 2585136675, 2807105250897, 4221456117363365, 8516994770090547979, 22470883218081146186209, 75883288444204588922998674, 322040154704144697047052726990

Offset: 0

N. J. A. Sloane

			a(0)=1 because the null graph (with no vertices) is vacuously 5-regular and connected.

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).

Leonard Chidiebere Eze, Robert Jajcay, and Jorik Jooken, On (k,g)-Graphs without (g+1)-Cycles, arXiv:2411.19023 [math.CO], 2024. See p. 18.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
Markus Meringer, Tables of Regular Graphs
Markus Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
Eric Weisstein's World of Mathematics, Quintic Graph
Eric Weisstein's World of Mathematics, Regular Graph

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
5-regular simple graphs: this sequence (connected), A165655 (disconnected), A165626 (not necessarily connected).
Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), this sequence (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 5-regular simple graphs with girth at least g: this sequence (g=3), A058275 (g=4), A205295 (g=5).
Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), A184954 (g=4), A184955 (g=5).
Connected 5-regular graphs: A129430 (loops and multiple edges allowed), A129419 (no loops but multiple edges allowed), this sequence (no loops nor multiple edges). (End)

a(n) = A184953(n) + A058275(n).
a(n) = A165626(n) - A165655(n).
Inverse Euler transform of A165626.

By running M. Meringer's GENREG for about 2 processor years at U. Newcastle, a(9) was found by Jason Kimberley, Nov 24 2009

a(10)-a(14) from Andrew Howroyd, Mar 10 2020

A014372 Number of trivalent connected simple graphs with 2n nodes and girth at least 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 9, 49, 455, 5783, 90938, 1620479, 31478584, 656783890, 14621871204, 345975648562

Offset: 0

N. J. A. Sloane

The null graph on 0 vertices is vacuously connected and 3-regular; since it is acyclic, it has infinite girth. - Jason Kimberley, Jan 29 2011

Brendan McKay has observed that a(13) = 31478584 is output by genreg, minibaum, and snarkhunter, but Meringer's table currently has a(13) = 31478582. - Jason Kimberley, May 17 2017

CRC Handbook of Combinatorial Designs, 1996, p. 647.

G. Brinkmann, J. Goedgebeur and B. D. McKay, Generation of cubic graphs, Discr. Math. Theor. Comp. Sci. 13 (2) (2011) 69-80.
House of Graphs, Cubic graphs.
Jason Kimberley, Connected regular graphs with girth at least 5
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.

Contribution from Jason Kimberley, 2010, 2011, and 2012: (Start)
3-regular simple graphs with girth at least 5: this sequence (connected), A185235 (disconnected), A185335 (not necessarily connected).
Connected k-regular simple graphs with girth at least 5: A186725 (all k), A186715 (triangle); A185115 (k=2), this sequence (k=3), A058343 (k=4), A205295 (g=5).
Connected 3-regular simple graphs with girth at least g: A185131 (triangle); A002851 (g=3), A014371 (g=4), this sequence (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7). (End)

Terms a(15) and a(16) appended, from running Meringer's GENREG for 28.7 and 715.2 processor days at U. Ncle., by Jason Kimberley, Jun 28 2010.

A014378 Number of connected regular graphs of degree 8 with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848873, 214755319657939505395, 187549729101764460261498, 186685399408147545744203815, 210977245260028917322933154987

Offset: 0

N. J. A. Sloane

Since the nontrivial 8-regular graph with the least number of vertices is K_9, there are no disconnected 8-regular graphs with less than 18 vertices. Thus for n<18 this sequence is identical to A180260. - Jason Kimberley, Sep 25 2009 and Feb 10 2011

			a(0)=1 because the null graph (with no vertices) is vacuously 8-regular and connected.

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.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Octic Graph
Eric Weisstein's World of Mathematics, Regular Graph

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
8-regular simple graphs: this sequence (connected), A165878 (disconnected), A180260 (not necessarily connected).
Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), this sequence (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
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: A184980 (triangle); chosen g: A184983 (g=3). (End)

a(n) = A184983(n) + A181154(n).
a(n) = A180260(n) + A165878(n).
This sequence is the inverse Euler transformation of A180260.

Using the symmetry of A051031, a(15) and a(16) were appended by Jason Kimberley, Sep 25 2009

a(17)-a(22) from Andrew Howroyd, Mar 13 2020

A006822 Number of connected regular graphs of degree 6 (or sextic graphs) with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7849, 367860, 21609300, 1470293675, 113314233808, 9799685588936, 945095823831036, 101114579937187980, 11945375659139626688, 1551593789610509806552, 220716215902792573134799, 34259321384370620122314325, 5782740798229825207562109439

Offset: 0

N. J. A. Sloane

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.
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 at least g
M. Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Regular Graph
Eric Weisstein's World of Mathematics, Sextic Graph

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
6-regular simple graphs: this sequence (connected), A165656 (disconnected), A165627 (not necessarily connected).
Connected regular graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), this sequence (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 6-regular simple graphs with girth at least g: this sequence (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4). (End)

a(n) = A184963(n) + A058276(n).
a(n) = A165627(n) - A165656(n).
This sequence is the inverse Euler transformation of A165627.

a(16) and a(17) appended, from running M. Meringer's GENREG at U. Newcastle for 41 processor days and 3.5 processor years, by Jason Kimberley, Sep 04 2009 and Nov 13 2009.

Terms a(18)-a(24), due to the extension of A165627 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020

A014374 Number of trivalent connected simple graphs with 2n nodes and girth at least 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 32, 385, 7574, 181227, 4624501, 122090544, 3328929954, 93990692595, 2754222605376

Offset: 0

N. J. A. Sloane

The null graph on 0 vertices is vacuously connected and 3-regular; since it is acyclic, it has infinite girth. [Jason Kimberley, Jan 29 2011]

CRC Handbook of Combinatorial Designs, 1996, p. 647.
M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146. [Jason Kimberley, Jan 29 2011]

House of Graphs, Cubic graphs
Jason Kimberley, Connected regular graphs with girth at least 6
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_, Jan 29 2011]

From Jason Kimberley, May 18 2010 and Jan 29 2011: (Start)
Connected k-regular simple graphs with girth at least 6: A186726 (any k), A186716 (triangle); specified degree k: A185116 (k=2), this sequence (k=3), A058348 (k=4).
Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), this sequence (g=6), A014375 (g=7), A014376 (g=8).
Connected 3-regular 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). (End)

Terms a(16) and a(17) appended, from running Meringer's GENREG for 18.6 and 530 processor days at U. Ncle., by Jason Kimberley on May 18 2010

Term a(18) from House of Graphs via Jason Kimberley, May 21 2017

cp's OEIS Frontend

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

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A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

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A068934 Triangular array C(n, r) = number of connected r-regular graphs with n nodes, 0 <= r < n.

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A005177 Number of connected regular graphs with n nodes.

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A014371 Number of trivalent connected simple graphs with 2*n nodes and girth at least 4.

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A006821 Number of connected regular graphs of degree 5 (or quintic graphs) with 2n nodes.

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A014372 Number of trivalent connected simple graphs with 2n nodes and girth at least 5.

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A014378 Number of connected regular graphs of degree 8 with n nodes.

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