A129437 -id:A129437 - cp's OEIS frontend

A129427 Number of isomorphism classes of 3-regular multigraphs of order 2n, loops allowed.

1, 2, 8, 31, 140, 722, 4439, 32654, 289519, 3054067, 37584620, 527968286, 8308434931, 144345554051, 2738280739075, 56245013793246, 1242596591479816, 29366532494796900, 739033832149588904, 19726887762569763453

Offset: 0

Brendan McKay, Apr 15 2007

nonn

a(1)..a(11) computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.

R. de Mello Koch, S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007
P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.
R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436. - _Jason Kimberley_, Sep 17 2009

Column k=3 of A167625.

Cf. A005967 (connected, inv. Euler trans.), A129416, A129429, A129431, A129433, A129435, A129437, A005638.

h = SymmetricFunctions(QQ).homogeneous()
def A129427(n):
    X = h([2*n]).plethysm(h([3]))
    Y = h([3*n]).plethysm(h([2]))
    return X.scalar(Y)
# Bruce Westbury, Aug 16 2013

a(n)=N\{S_{2n}[S_3] * S_{3n}[S_2]\}. - Jason Kimberley, Sep 17 2009

Using equation (5.8) of Read 1959, new terms a(12) and a(13) were computed in MAGMA by Jason Kimberley, Sep 17 2009
Further terms a(14)-a(16) also computed by Jason Kimberley, announced Nov 09 2009
Formula corrected from n vertices to 2n vertices by Jason Kimberley, Nov 09 2009
Added a(0). - N. J. A. Sloane, Aug 26 2013
a(17)-a(19) from Sean A. Irvine, Oct 29 2016

A129429 Number of isomorphism classes of 4-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 3, 7, 20, 56, 187, 654, 2705, 12587, 67902, 417065, 2897432, 22382255, 189930004, 1750561160, 17380043136, 184653542135, 2088649831822, 25046462480066, 317295911519901, 4233450347175663, 59329632953577985, 871281036897298464

Offset: 1

Brendan McKay, Apr 15 2007

nonn

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

Equation (5.8) of Read's paper tells us a(n) = N {S_n[S_4] * S_{2n}[S_2]}, where we are working with cycle index polynomials. - Jason Kimberley, Oct 05 2009

Andrew Howroyd, Table of n, a(n) for n = 1..40
R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007
R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436.

Column k=4 of A167625.

Cf. A085549 (connected), A129418, A129427, A129431, A129433, A129435, A129437.

Euler transform of A085549. - Andrew Howroyd, Mar 15 2020

Using equation (5.8) of Read's paper, new terms a(17)-a(19) were computed in MAGMA by Jason Kimberley, Oct 05 2009

Four more terms a(20)-a(23) also computed by Jason Kimberley, Nov 09 2009

A167625 Square array T(n,k), read by upward antidiagonals, counting isomorphism classes of k-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 3, 2, 1, 1, 0, 5, 0, 3, 0, 1, 1, 7, 8, 7, 3, 1, 1, 0, 11, 0, 20, 0, 4, 0, 1, 1, 15, 31, 56, 32, 13, 4, 1, 1, 0, 22, 0, 187, 0, 66, 0, 5, 0, 1, 1, 30, 140, 654, 727, 384, 101, 22, 5, 1, 1, 0, 42, 0, 2705, 0, 3369, 0, 181, 0, 6, 0, 1, 1, 56, 722, 12587, 42703

Offset: 1

Jason Kimberley, Nov 07 2009

The number of vertices n is positive; valency k is nonnegative.
Each loop contributes two to the valency of its vertex.
The antidiagonal having coordinate sum t=n+k is read from T(t,0) to T(1,t-1).
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 A333467. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 23 2020

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  1 | 1 0  1   0    1     0      1       0 ...
  2 | 1 1  2   2    3     3      4       4 ...
  3 | 1 0  3   0    7     0     13       0 ...
  4 | 1 1  5   8   20    32     66     101 ...
  5 | 1 0  7   0   56     0    384       0 ...
  6 | 1 1 11  31  187   727   3369   12782 ...
  7 | 1 0 15   0  654     0  40365       0 ...
  8 | 1 1 22 140 2705 42703 675368 8584767 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..378 (27 antidiagonals, first 19 antidiagonals from Jason Kimberley)
J. S. Kimberley, Table in user subpage of wiki.
R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436.

Column sequences: A000012 (k=0), A059841 (k=1), A000041 (k=2), A129427 (k=3), A129429 (k=4), A129431 (k=5), A129433 (k=6), A129435 (k=7), A129437 (k=8).

Cf. A333330 (loopless), A333397 (connected), A333467 (labeled).

T(n,k) = N\{S_n[S_k] * S_{nk/2}[S_2]\}.

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

A129431 Number of isomorphism classes of 5-regular multigraphs of order 2n, loops allowed.

Original entry on oeis.org

3, 32, 727, 42703, 5988679, 1639714425, 757559332934, 541249158493444, 564262722366313620, 822164422526588575949, 1618567795242262158194706, 4188563149202582371775198174, 13926836449718334345103644635724, 58360974360850795591633858610837541

Offset: 1

Brendan McKay, Apr 15 2007

nonn

First seven terms were 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_, Oct 05 2009]

Column k=5 of A167625.

Cf. A129430, A129420, A129427, A129429, A129433, A129435, A129437.

a(n) = N\{S_{2n}[S_5] * S_{5n}[S_2]\}. - Jason Kimberley, Oct 05 2009

Euler transform of A129430. - Andrew Howroyd, Mar 15 2020

Using equation (5.8) of Read 1959, new terms a(8)-a(10) were computed in MAGMA during 2009 by Jason Kimberley, Dec 22 2010

a(11)-a(14) from Andrew Howroyd, Mar 21 2020

A129433 Number of isomorphism classes of 6-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 4, 13, 66, 384, 3369, 40365, 675368, 14843787, 412444439, 14024069358, 570883006810, 27368160642418, 1525274952640101, 97766102550401217, 7141331463919539567, 589726897233157151109, 54669618693986578729541, 5653861386413841612952683, 648651469510725736002532451

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_, Oct 05 2009]

Column k=6 of A167625.

Cf. A129432, A129422, A129427, A129429, A129431, A129435, A129437.

a(n) = N\{S_n[S_6] * S_{3n}[S_2]\}. [Equation (5.8) of Read 1959]

Euler transform of A129432. - Andrew Howroyd, Mar 15 2020

Using the formula, new terms a(13), Oct 05 2009, and a(14)-a(16), Nov 09 2009, were computed in MAGMA by Jason Kimberley

a(17)-a(20) 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

A129436 Number of isomorphism classes of connected 8-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 4, 17, 149, 1934, 44821, 1631629, 89057367, 6849428873, 713780361312, 97876276145119, 17259548258350637, 3840154740252625874, 1060662127742505706789, 358584059544008234423217, 146560585570176100774010071, 71630591614693085251230481320, 41456445821273701849195905028292

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 A333397.

Cf. A129437, A129425, A005967, A085549, A129430, A129432, A129434.

Inverse Euler transform of A129437. - Andrew Howroyd, Mar 19 2020

a(11)-a(18) added by 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

A129427 Number of isomorphism classes of 3-regular multigraphs of order 2n, loops allowed.

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A129429 Number of isomorphism classes of 4-regular multigraphs of order n, loops allowed.

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A167625 Square array T(n,k), read by upward antidiagonals, counting isomorphism classes of k-regular multigraphs of order n, loops allowed.

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

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A129431 Number of isomorphism classes of 5-regular multigraphs of order 2n, loops allowed.

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A129433 Number of isomorphism classes of 6-regular multigraphs of order n, loops allowed.

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

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A129436 Number of isomorphism classes of connected 8-regular multigraphs of order n, loops allowed.

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

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