A129426 -id:A129426 - cp's OEIS frontend

A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 3, 2, 1, 1, 1, 0, 1, 0, 4, 0, 4, 0, 1, 1, 0, 1, 1, 5, 7, 9, 4, 1, 1, 1, 0, 1, 0, 7, 0, 24, 0, 7, 0, 1, 1, 0, 1, 1, 8, 16, 54, 60, 32, 8, 1, 1, 1, 0, 1, 0, 10, 0, 128, 0, 240, 0, 12, 0, 1, 1, 0, 1, 1, 12, 37, 271, 955, 1753, 930, 135, 14, 1, 1

Offset: 0

Andrew Howroyd, Mar 15 2020

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A333351. Burnside's lemma can be used to extend this method to the unlabeled case.

			Array begins:
=================================================
n\k | 0 1 2  3   4    5      6     7        8
----+--------------------------------------------
  0 | 1 1 1  1   1    1      1     1        1 ...
  1 | 1 0 0  0   0    0      0     0        0 ...
  2 | 1 1 1  1   1    1      1     1        1 ...
  3 | 1 0 1  0   1    0      1     0        1 ...
  4 | 1 1 2  3   4    5      7     8       10 ...
  5 | 1 0 2  0   7    0     16     0       37 ...
  6 | 1 1 4  9  24   54    128   271      582 ...
  7 | 1 0 4  0  60    0    955     0    12511 ...
  8 | 1 1 7 32 240 1753  13467 90913   543779 ...
  9 | 1 0 8  0 930    0 253373     0 35255015 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..350

Columns k=0..8 are (with interspersed 0's for odd k): A000012, A000012, A002865, A129416, A129418, A129420, A129422, A129424, A129426.
Row n=4 is A001399.
Cf. A051031 (simple graphs), A167625 (with loops), A192517 (not necessarily regular), A328682 (connected), A333351 (labeled nodes).

A129416 Number of isomorphism classes of 3-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 3, 9, 32, 135, 709, 4637, 38374, 391473, 4764778, 66913591, 1056886475, 18446472265, 351482430368, 7247888726269, 160671989129665, 3808499268504548, 96094161981827499, 2570930535917564366, 72688753062897675445

Offset: 1

Brendan McKay, Apr 15 2007

nonn

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

L. Travis, Graphical Enumeration: A Species-Theoretic Approach, arXiv:math/9811127 [math.CO], 1998.; Ph.D. thesis, Brandeis University, 1999, Section 4.3.

Column k=3 of A333330.

Cf. A000421 (connected, inv. Eul. trans.), A129427, A129418, A129420, A129422, A129424, A129426.

Euler transform of A000421.

a(13)-a(20) from Andrew Howroyd, Mar 19 2020

A129425 Number of isomorphism classes of connected 8-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 9, 36, 571, 12465, 543116, 35241608, 3230417239, 397514307014, 63830872225605, 13080448625309965, 3358687593761378470, 1063838242661288090062, 410057057694777406364151, 190064879184725871853627854, 104825763290631293396894238206

Offset: 1

Brendan McKay, Apr 15 2007

nonn

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Brendan McKay and Adolfo Piperno, Nauty and Traces

Column k=8 of A328682.

Cf. A129426, A129436, A000421, A129417, A129419, A129421, A129423, A328682.

geng -c -d1 ${n} -q | multig -r8 -u # Natan Arie Consigli, Jun 05 2017

Inverse Euler transform of A129426. - Andrew Howroyd, Mar 17 2020

Deleted a(0) and a(1). - N. J. A. Sloane, Jan 11 2020

a(1)=0 prepended and a(12)-a(18) from Andrew Howroyd, Mar 17 2020

A129437 Number of isomorphism classes of 8-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 5, 22, 181, 2183, 47773, 1689841, 90972682, 6948008975, 721121538707, 98626660242232, 17361164959413148, 3857920886847174328, 1064590883337392451345, 359664321908847682542521, 146924560811224485478212220, 71778969194383385732028947007, 41528817306107485906955040499799

Offset: 1

Brendan McKay, Apr 15 2007

nonn

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436. [From _Jason Kimberley_, Nov 09 2009]

Column k=8 of A167625.

Cf. A129436, A129426, A129427, A129429, A129431, A129433, A129435.

a(n)=N\{S_n[S_8] * S_{4n}[S_2]\}. - Jason Kimberley, Nov 09 2009

Euler transform of A129436. - Andrew Howroyd, Mar 15 2020

Using equation (5.8) of Read 1959, McKay's terms verified by, and new term a(11) was computed by Jason Kimberley, Nov 09 2009
a(12)=N{S_12[S_8]*S_48[S_2]} was computed in MAGMA, on one processor of ARCSgrid at UNcle, using 17 GB virtual memory, over 49 real days, with 36 days processor time, by Jason Kimberley, Nov 29 2009
a(13)-a(18) from Andrew Howroyd, Mar 21 2020

A129418 Number of isomorphism classes of 4-regular loopless multigraphs of order n.

Original entry on oeis.org

1, 0, 1, 1, 4, 7, 24, 60, 240, 930, 4701, 26637, 178569, 1339529, 11187064, 101871881, 1002594996, 10574095327, 118850827173, 1417140114336, 17860018997346, 237160827107408, 3309078044759285, 48396906463199522, 740331404753448181, 11821525310570525197

Offset: 0

Brendan McKay, Apr 15 2007

nonn

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Also number of carbon allotropes satisfying the octet rule, excluding stereoisomers. - Natan Arie Consigli, Jun 06 2017

Column k=4 of A333330.

Cf. A129417, A129429, A129416, A129420, A129422, A129424, A129426.

geng -d1 ${n} -q | multig -r4 -u # Natan Arie Consigli, Jun 06 2017

Euler transform of A129417. - Andrew Howroyd, Mar 14 2020

a(0)-a(1) by Natan Arie Consigli, Jun 06 2017

a(18)-a(25) from Andrew Howroyd, Mar 17 2020

A129420 Number of isomorphism classes of 5-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 5, 54, 1753, 189341, 46935710, 20494522535, 14041749098602, 14155266802426836, 20061744131278672638, 38587417589460488631726, 97900485588988429336271590, 320012505326477694925887757141, 1321269556386383657509085883067690, 6775074159053505093089897813890701467

Offset: 1

Brendan McKay, Apr 15 2007

nonn

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Column k=5 of A333330.

Cf. A129419, A129431, A129416, A129418, A129422, A129424, A129426.

Euler transform of A129419. - Andrew Howroyd, Mar 17 2020

a(8)-a(15) from Andrew Howroyd, Mar 21 2020

A129422 Number of isomorphism classes of 6-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 7, 16, 128, 955, 13467, 253373, 6466074, 205749149, 7943313377, 363853255012, 19485170158346, 1205488841884007, 85308028236495340, 6846462326434510551, 618498122199399056707, 62478078728492272712838, 7015617595855429187696753

Offset: 1

Brendan McKay, Apr 15 2007

nonn

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Column k=6 of A333330.

Cf. A129421, A129433, A129416, A129418, A129420, A129424, A129426.

Euler transform of A129421. - Andrew Howroyd, Mar 14 2020

a(1)=0 prepended and a(14)-a(20) from Andrew Howroyd, Mar 17 2020

A129424 Number of isomorphism classes of 7-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 8, 271, 90913, 165134083, 861889423083, 10352781874872853, 253226977418079364874, 11542716741789258789499297, 915461016322959135074642849828

Offset: 1

Brendan McKay, Apr 15 2007

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Column k=7 of A333330.

Cf. A129423, A129435, A129416, A129418, A129420, A129422, A129426.

Euler transform of A129423. - Andrew Howroyd, Mar 21 2020

a(7)-a(10) from Andrew Howroyd, Mar 21 2020

A180260 Number of not necessarily connected 8-regular simple graphs on n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848874, 214755319657939505396, 187549729101764460261505, 186685399408147545744203915, 210977245260028917322933165888

Offset: 0

Jason Kimberley, Jan 17 2011

The Euler transformation currently does nothing: for n < 18, a(n) = A014378(n).

			The a(0)=1 graph is K_0 (vacuously 8-regular).
The a(9)=1 graph is K_9.

Eric Weisstein's World of Mathematics, Octic Graph

8-regular simple graphs: A014378 (connected), A165878 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A000012 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), this sequence (k=8).
8-regular not necessarily connected graphs: this sequence (simple graphs), A129437 (multigraphs with loops allowed), A129426 (multigraphs with loops forbidden).

A014378 = Cases[Import["https://oeis.org/A014378/b014378.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014378] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014378.

a(17)-a(22) from Andrew Howroyd, Mar 08 2020

cp's OEIS Frontend

A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

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A129416 Number of isomorphism classes of 3-regular loopless multigraphs of order 2n.

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A129425 Number of isomorphism classes of connected 8-regular loopless multigraphs of order n.

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nauty

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A129437 Number of isomorphism classes of 8-regular multigraphs of order n, loops allowed.

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A129418 Number of isomorphism classes of 4-regular loopless multigraphs of order n.

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nauty

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A129420 Number of isomorphism classes of 5-regular loopless multigraphs of order 2n.

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A129422 Number of isomorphism classes of 6-regular loopless multigraphs of order n.

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A129424 Number of isomorphism classes of 7-regular loopless multigraphs of order 2n.

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A180260 Number of not necessarily connected 8-regular simple graphs on n vertices.

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