A328682 -id:A328682 - cp's OEIS frontend

A000421 Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.

1, 2, 6, 20, 91, 509, 3608, 31856, 340416, 4269971, 61133757, 978098997, 17228295555, 330552900516, 6853905618223, 152626436936272, 3631575281503404, 91928898608055819, 2466448432564961852, 69907637101781318907

Offset: 1

N. J. A. Sloane

a(n) is also the number of isomorphism classes of connected 3-regular simple graphs of order 2n with possibly loops. - Nico Van Cleemput, Jun 04 2014

There are no graphs of order 2n+1 satisfying the condition above. - Natan Arie Consigli, Dec 20 2019

			From _Natan Arie Consigli_, Dec 20 2019: (Start)
a(1) = 1: with two nodes the only viable option is the triple edged path multigraph.
a(2) = 4: with four nodes we have two cases: the tetrahedral graph and the square graph with single and double edges on opposite sides.
(End)

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92 [gives incorrect a(6)].
CRC Handbook of Combinatorial Designs, 1996, p. 651 [or: 2006, table 4.40].

Jan-Peter Börnsen, Anton E. M. van de Ven, Tangent Developable Orbit Space of an Octupole, arXiv:1807.04817 [hep-th], 2018.
G. Brinkmann, N. Van Cleemput, T. Pisanski, Generation of various classes of trivalent graphs, Theoretical Computer Science 502, 2013, pp.16-29.
R. J. Mathar, Cubic multigraphs A000421
Brendan McKay and others, Nauty Traces

Column k=3 of A328682 (table of k-regular n-node multigraphs).

Cf. A129416, A005967 (loops allowed), A129417, A129419, A129421, A129423, A129425, A002851 (no multiedges).

for n in {1..10}; do geng -cqD3 $[2*$n] | multig -ur3; done # Sean A. Irvine, Sep 24 2015

Inverse Euler transform of A129416. - Andrew Howroyd, Mar 19 2020

More terms from Brendan McKay, Apr 15 2007

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

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

Original entry on oeis.org

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).

A129417 Number of isomorphism classes of connected 4-regular loopless multigraphs of order n.

Original entry on oeis.org

1, 0, 1, 1, 3, 6, 19, 50, 204, 832, 4330, 25227, 171886, 1303725, 10959478, 100230117, 989280132, 10455393155, 117701173970, 1405165683359, 17726785643045, 235585551038117, 3289367315407521, 48136794098893837, 736721822918719557, 11768987500655142988

Offset: 0

Brendan McKay, Apr 15 2007

nonn

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

Obtained from A129418 by an inverse Euler transform. - R. J. Mathar, Mar 09 2019

Brendan McKay and Adolfo Piperno, Nauty and Traces

Column k=4 of A328682.

Cf. A129418, A085549, A000421, A129419, A129421, A129423, A129425, A328682.

geng -c -d1 ${n} -q | multig -r4 -u # Natan Arie Consigli, Jun 05 2017

a(0)-a(1) prepended by Natan Arie Consigli, Jun 05 2017

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

A129419 Number of isomorphism classes of connected 5-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 4, 49, 1689, 187392, 46738368, 20446754006, 14021056991357, 14141140657400321, 20047531681346319557, 38567298550226625579671, 97861817259606311572409609, 319914449561753621623849929222, 1320949150506412557504787822889933, 6773751604973857152218372443743552754

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=5 of A328682.

Cf. A129420, A129430, A000421, A129417, A129421, A129423, A129425, A328682.

geng -c -d1 $[2*$n] -q | multig -r5 -u # Natan Arie Consigli, Dec 20 2019

Inverse Euler transform of A129420. - Andrew Howroyd, Mar 17 2020

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

A129421 Number of isomorphism classes of connected 6-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 6, 15, 120, 933, 13303, 252207, 6450828, 205475039, 7936493756, 363639228194, 19476976825809, 1205115679461426, 85288127619421544, 6845235025444882069, 618411485467843477405, 62471139399366989007575, 7014991719815977343879171

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=6 of A328682.

Cf. A129422, A129432, A000421, A129417, A129419, A129423, A129425, A328682.

geng -c -d1 ${n} -q | multig -r6 -u # Natan Arie Consigli, Jun 05 2017

Inverse Euler transform of A129422. - Andrew Howroyd, Mar 17 2020

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

A333397 Array read by antidiagonals: T(n,k) is the number of connected k-regular multigraphs on n unlabeled nodes, loops allowed, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 3, 4, 5, 1, 0, 0, 1, 0, 3, 0, 10, 0, 1, 0, 0, 1, 1, 4, 9, 26, 28, 17, 1, 0, 0, 1, 0, 4, 0, 47, 0, 97, 0, 1, 0, 0, 1, 1, 5, 17, 91, 291, 639, 359, 71, 1, 0, 0, 1, 0, 5, 0, 149, 0, 2789, 0, 1635, 0, 1, 0, 0

Offset: 0

Andrew Howroyd, Mar 18 2020

This sequence can be derived from A167625 by inverse Euler transform.

			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 1  0    1     0        1       0          1 ...
  2 | 0 1 1  2    2     3        3       4          4 ...
  3 | 0 0 1  0    4     0        9       0         17 ...
  4 | 0 0 1  5   10    26       47      91        149 ...
  5 | 0 0 1  0   28     0      291       0       1934 ...
  6 | 0 0 1 17   97   639     2789   12398      44821 ...
  7 | 0 0 1  0  359     0    35646       0    1631629 ...
  8 | 0 0 1 71 1635 40264   622457 8530044   89057367 ...
  9 | 0 0 1  0 8296     0 14019433       0 6849428873 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..405 (first 28 antidiagonals)

Columns k=3..8 (with interspersed 0's for odd k) are: A005967, A085549, A129430, A129432, A129434, A129436.

Cf. A167625 (not necessarily connected), A322115 (not necessarily regular), A328682 (loopless), A333330.

Column k is the inverse Euler transform of column k of A167625.

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

A334546 Array read by antidiagonals: T(n,k) is the number of unlabeled connected loopless multigraphs with n nodes of degree k or less.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 3, 2, 0, 0, 1, 1, 4, 4, 2, 0, 0, 1, 1, 5, 9, 12, 2, 0, 0, 1, 1, 6, 14, 37, 22, 2, 0, 0, 1, 1, 7, 23, 93, 146, 68, 2, 0, 0, 1, 1, 8, 32, 203, 602, 772, 166, 2, 0, 0, 1, 1, 9, 46, 399, 2126, 5847, 4449, 534, 2, 0, 0

Offset: 0

Andrew Howroyd, May 05 2020

This sequence may be derived from A333893 by inverse Euler transform.

			Array begins:
==============================================
n\k | 0 1 2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1 1   1    1     1      1       1 ...
  1 | 1 1 1   1    1     1      1       1 ...
  2 | 0 1 2   3    4     5      6       7 ...
  3 | 0 0 2   4    9    14     23      32 ...
  4 | 0 0 2  12   37    93    203     399 ...
  5 | 0 0 2  22  146   602   2126    6308 ...
  6 | 0 0 2  68  772  5847  34126  164965 ...
  7 | 0 0 2 166 4449 66289 716141 6021463 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals 0..26)

Columns k=3..5 are A243391, A289157, A334547.
Main diagonal is A334546.
Cf. A289987, A328682 (regular), A333893 (not necessarily connected).

Column k is the inverse Euler transform of column k of A333893.

A324218 Number of connected n-regular loopless multigraphs on six unlabeled nodes.

Original entry on oeis.org

0, 0, 1, 6, 19, 49, 120, 263, 571, 1149, 2259, 4218, 7679, 13437, 22952, 38013, 61580, 97309, 150838, 229045, 342048, 502066, 726318, 1035482, 1457677, 2026369, 2785873, 3788494, 5101847, 6805192, 8998964, 11799997, 15353938, 19829063, 25431994, 32400113, 41021075, 51623423

Offset: 0

Natan Arie Consigli, Feb 25 2019

nonn

Multigraphs are loopless.

Terms computed using 'nauty and Traces' (see the link).

Brendan McKay and Adolfo Piperno, nauty and Traces

Row n=6 of A328682.

Cf. A324221, A324217.

for ((n=0;n<50;n++)); do geng -c -d1 6 -q | multig -r${n} -u; done

cp's OEIS Frontend

A000421 Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.

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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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A129417 Number of isomorphism classes of connected 4-regular loopless multigraphs of order n.

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A129419 Number of isomorphism classes of connected 5-regular loopless multigraphs of order 2n.

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A129421 Number of isomorphism classes of connected 6-regular loopless multigraphs of order n.

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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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A333397 Array read by antidiagonals: T(n,k) is the number of connected k-regular multigraphs on n unlabeled nodes, loops allowed, n >= 0, k >= 0.

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

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