A129424 -id:A129424 - 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

A129426 Number of isomorphism classes of 8-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 10, 37, 582, 12511, 543779, 35255015, 3230979297, 397550237967, 63834143947661, 13080849749829233, 3358751856150607392, 1063851391062768324862, 410060430118305494628648, 190065946515113295597969794, 104826174445642584491349328181

Offset: 1

Brendan McKay, Apr 15 2007

nonn

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

Column k=8 of A333330.

Cf. A129425, A129437, A129416, A129418, A129420, A129422, A129424.

Euler transform of A129425. - Andrew Howroyd, Mar 17 2020

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

A129423 Number of isomorphism classes of connected 7-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 7, 263, 90614, 165041329, 861723619902, 10351918806321621, 253216618556625008961, 11542463442106815907796586, 915449471830886733265105097578

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=7 of A328682.

Cf. A129424, A129434, A000421, A129417, A129419, A129421, A129425, A328682.

geng -c -d1 $[2*$n] -q | multig -r7 -u # Natan Arie Consigli, Dec 20 2019

Inverse Euler transform of A129424. - Andrew Howroyd, Mar 21 2020

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

A129435 Number of isomorphism classes of 7-regular multigraphs of order 2n, loops allowed.

Original entry on oeis.org

4, 101, 12782, 8584767, 20104116089, 122644465172798, 1658339657066189475, 44564623565972592394826, 2193250056291167380214634054, 185389574171283940222059091478222

Offset: 1

Brendan McKay, Apr 15 2007

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.

Column k=7 of A167625.

Cf. A129434, A129424, A129427, A129429, A129431, A129433, A129437.

a(n) = N\{S_{2n}[S_7] * S_{7n}[S_2]\}.

Euler transform of A129434. - Andrew Howroyd, Mar 15 2020

Using equation (5.8) of Read 1959, McKay's terms were verified by, and new term a(6) was computed by Jason Kimberley, Nov 09 2009

a(7)-a(10) from Andrew Howroyd, Mar 21 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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A129426 Number of isomorphism classes of 8-regular loopless multigraphs of order n.

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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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A129423 Number of isomorphism classes of connected 7-regular loopless multigraphs of order 2n.

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nauty

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A129435 Number of isomorphism classes of 7-regular multigraphs of order 2n, loops allowed.

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