A033483 -id:A033483 - 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

A165652 Number of disconnected 2-regular graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 8, 9, 12, 16, 20, 24, 32, 38, 48, 59, 72, 87, 109, 129, 157, 190, 229, 272, 330, 390, 467, 555, 659, 778, 926, 1086, 1283, 1509, 1774, 2074, 2437, 2841, 3322, 3871, 4509, 5236, 6094, 7055, 8181, 9464, 10944, 12624, 14577, 16778, 19322, 22209

Offset: 0

Jason Kimberley, Sep 28 2009

a(n) is also the number of partitions of n such that each part i satisfies 2

For n>=2, it appears that a(n+1) is the number of (1,0)-separable partitions of n, as defined at A239482. For example, the four (1,0)-separable partitions of 9 are 621, 531, 441, 31212, corresponding to a(10) = 4. - Clark Kimberling, Mar 21 2014.

			The a(6)=1 graph is C_3+C_3. The a(7)=1 graph is C_3+C_4. The a(8)=2 graphs are C_3+C_5, C_4+C_4. The a(9)=3 graphs are 3C_3, C_3+C_6, C_4+C_5.

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

2-regular simple graphs: A179184 (connected), this sequence (disconnected), A008483 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A157928 (k=0), A157928 (k=1), this sequence (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8).
Disconnected 2-regular simple graphs with girth at least g: this sequence (g=3), A185224 (g=4), A185225 (g=5), A185226 (g=6), A185227 (g=7), A185228 (g=8), A185229 (g=9).
Cf. A239482.

p := NumberOfPartitions; a := func< n | n lt 3 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3) - 1 >;

a = A008483 - A179184 = Euler_tranformation(A179184) - A179184.
For n > 2, since there is exactly one connected 2-regular graph on n vertices (the n cycle C_n) then a(n) = A008483(n) - 1.
(A008483(n) is also the number of not necessarily connected 2-regular graphs on n vertices.)
Column D(n, 2) in the triangle A068933.

A033301 Number of 4-valent (or quartic) graphs with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 60, 266, 1547, 10786, 88193, 805579, 8037796, 86223660, 985883873, 11946592242, 152808993767, 2056701139136, 29051369533596, 429669276147047, 6640178380127244, 107026751932268789, 1796103830404560857, 31334029441145918974, 567437704731717802783

Offset: 0

Ronald C. Read

Because the triangle A051031 is symmetric, a(n) is also the number of (n-5)-regular graphs on n vertices. - Jason Kimberley, Sep 22 2009

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

M. Meringer, Tables of Regular Graphs
M. Meringer, Erzeugung Regulärer Graphen, Diploma thesis, University of Bayreuth, January 1996. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010]
N. J. A. Sloane, Transforms
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, Quartic Graph

4-regular simple graphs: A006820 (connected), A033483 (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), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).

A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] (* Jean-François Alcover, Nov 26 2019, updated Mar 17 2020 *)

Euler transform of A006820. - Martin Fuller, Dec 04 2006

a(16) from Axel Kohnert (kohnert(AT)uni-bayreuth.de), Jul 24 2003
a(17)-a(19) from Jason Kimberley, Sep 12 2009
a(20)-a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
a(22) from Jason Kimberley, Oct 15 2011
a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 08 2020

A185244 Number of disconnected 4-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 15, 35, 247, 1692, 17409, 197924, 2492824, 33117880, 461597957, 6709514218, 101153412903, 1597440868898

Offset: 0

Jason Kimberley, Feb 22 2011

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

4-regular simple graphs with girth at least 4: A033886 (connected), this sequence (disconnected), A185344 (not necessarily connected).
Disconnected 4-regular simple graphs with girth at least g: A033483 (g=3), this sequence (g=4), A185245 (g=5), A185246 (g=6).
Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), this sequence (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

a(n) = A185344(n) - A033886(n) = Euler_transformation(A033886)(n) - A033886(n).

a(n) = A185044(n) + A185245(n).

a(31) appended by the author once A033886(23) was known, Nov 03 2011

a(31) corrected by the author, Jan 05 2013

A165653 Number of disconnected 3-regular (cubic) graphs on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 9, 31, 147, 809, 5855, 54477, 633057, 8724874, 137047391, 2391169355, 45626910415, 942659626031, 20937539944549, 497209670658529, 12566853576025106, 336749273734805530, 9534909974420181226

Offset: 0

Jason Kimberley, Sep 28 2009

Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Cubic Graph
Eric Weisstein's World of Mathematics, Disconnected Graph

3-regular simple graphs: A002851 (connected), this sequence (disconnected), A005638 (not necessarily connected).

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), this sequence (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A005638 = A@005638;
A002851 = A@002851;
a[n_] := A005638[[n + 1]] - A002851[[n + 1]];
a /@ Range[0, 20] (* Jean-François Alcover, Jan 21 2020 *)

a(n) = A005638(n) - A002851(n).

a(n) = A068933(2n, 3).

A165655 Number of disconnected 5-regular (quintic) graphs on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 66, 8029, 3484760, 2595985770, 2815099031417, 4230059694039460, 8529853839173455678, 22496718465713456081402, 75951258300080722467845995, 322269241532759484921710401976

Offset: 0

Jason Kimberley, Sep 28 2009

N. J. A. Sloane, Transforms
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Quintic Graph

5-regular simple graphs: A006821 (connected), this sequence (disconnected), A165626 (not necessarily connected).

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), this sequence (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

a = A165626 - A006821 = Euler_transformation(A006821) - A006821.

a(n)=A068933(2n,5).

Terms a(13)-a(17), due to the extension of A006821 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020

A165656 Number of disconnected 6-regular (sextic) graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 25, 297, 8199, 377004, 22014143, 1493574756, 114880777582, 9919463450855, 955388277929620, 102101882472479938, 12050526046888229845, 1563967741064673811531, 222318116370232302781485, 34486536277291555593662301, 5817920265098158804699762770

Offset: 0

Jason Kimberley, Sep 28 2009

N. J. A. Sloane, Transforms
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Regular Graph
Eric Weisstein's World of Mathematics, Sextic Graph

6-regular simple graphs: A006822 (connected), this sequence (disconnected), A165627 (not necessarily connected).

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), this sequence (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

a = A165627 - A006822 = Euler_transformation(A006822) - A006822.

a(n) = D(n, 6) in the triangle A068933.

Terms a(25) and beyond from Andrew Howroyd, May 20 2020

A165877 Number of disconnected 7-regular (septic) graphs on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 1562, 21617036, 733460349818, 42703733735064572, 4073409466378991404239, 613990076321940092226829047, 141518698937232822678583027258225

Offset: 0

Jason Kimberley, Sep 28 2009

N. J. A. Sloane, Transforms
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Septic Graph

7-regular simple graphs: A014377 (connected), this sequence(disconnected), A165628 (not necessarily connected).

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), this sequence (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

a = A165628 - A014377 = Euler_transformation(A014377) - A014377.

a(n)=D(2n, 7) in the triangle A068933.

a(13)-a(16) from Andrew Howroyd, May 20 2020

A165878 Number of disconnected 8-regular simple graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 100, 10901, 3470736, 1473822243, 734843169811, 423929978716908, 281768931380519766, 215039290728074333738, 187766225244288486398132, 186874272297562916477691894, 211165081721567703008217979077

Offset: 0

Jason Kimberley, Sep 29 2009

			The a(18)=1 graph is K_9+K_9.

Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Octic Graph

8-regular simple graphs: A014378 (connected), this sequence (disconnected), A180260 (not necessarily connected).

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), this sequence (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

a = A180260 - A014378 = Euler_transformation(A014378) - A014378.

a(n) = D(n, 8) in the triangle A068933.

Terms a(26) and beyond from Andrew Howroyd, May 20 2020

A185203 Number of disconnected 10-regular graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 11, 550, 806174, 2585947720, 9802278927562, 42709859521915286, 214798119408798346811, 1251607430636395979871600, 8463468717232507491862780325, 66406919318277846825588474735084

Offset: 0

Jason Kimberley, Jan 26 2012

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

10-regular simple graphs: A014382 (connected), this sequence (disconnected).

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), this sequence (k=10), A185213 (k=11).

Terms a(29) and beyond from Andrew Howroyd, May 20 2020

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cp's OEIS Frontend

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

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A165652 Number of disconnected 2-regular graphs on n vertices.

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A033301 Number of 4-valent (or quartic) graphs with n nodes.

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A185244 Number of disconnected 4-regular simple graphs on n vertices with girth at least 4.

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A165653 Number of disconnected 3-regular (cubic) graphs on 2n vertices.

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A165655 Number of disconnected 5-regular (quintic) graphs on 2n vertices.

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A165656 Number of disconnected 6-regular (sextic) graphs on n vertices.

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A165877 Number of disconnected 7-regular (septic) graphs on 2n vertices.

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A165878 Number of disconnected 8-regular simple graphs on n vertices.

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