A129427 -id:A129427 - cp's OEIS frontend

A003175 Almost certainly an erroneous version of A129427.

Original entry on oeis.org

1, 2, 8, 31, 139, 724

Offset: 0

N. J. A. Sloane, Jul 02 2015

dead

P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Cf. A129427, A005638.

A005967 Number of isomorphism classes of connected 3-regular multigraphs of order 2n, loops allowed.

Original entry on oeis.org

2, 5, 17, 71, 388, 2592, 21096, 204638, 2317172, 30024276, 437469859, 7067109598, 125184509147, 2410455693765, 50101933643655, 1117669367609605, 26629298567576331, 674793598023809924, 18119844622209998036

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of maximal cells in the moduli space of tropical curves of genus n+1; see Melody Chan (2012) reference. a(n) is also the number of maximally degenerate stable nodal algebraic curves of genus n+1, up to isomorphism, by the association of a stable nodal curve to its dual graph. - Harry Richman, Oct 23 2023

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Brinkmann, N. Van Cleemput, and T. Pisanski, Generation of various classes of trivalent graphs, Theor. Comput. Sci. 502 (2013) 16-29, Table 1 column LM.
Melody Chan, Combinatorics of the tropical Torelli map, Algebra Number Theory, 6 (2012), 1133-1169.
Melody Chan, Moduli Spaces of Curves: Classical and Tropical, Notices Amer. Math. Soc., 68 (2021), 1700-1713.
R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007 (D7).
R. J. Mathar, Cubic Multigraphs A005967
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019).
Brendan McKay, nauty software
Wikipedia, Moduli of algebraic curves.

Column k=3 of A333397.

Cf. A129427 (Euler transf.), A000421 (no loops), A085549, A129430, A129432, A129434, A129436.

Inverse Euler transform of A129427.

Checked by Brendan McKay, Apr 15 2007
Using sequence A129427, terms a(12)-a(16) were computed in GAP by Ignat Soroko, Apr 07 2010
a(17)-a(19) added by Andrew Howroyd, Mar 19 2020

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]\}.

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

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

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

A352175 The number of Feynman graphs in phi^3 theory with 2n vertices, 2 external legs.

Original entry on oeis.org

1, 5, 30, 186, 1276, 9828, 86279, 866474, 9924846, 128592118, 1864888539, 29950693288, 527584198445, 10109318656565, 209256249845854, 4651751087878667, 110501782280985273, 2792991694461152344, 74832356485576239136, 2118333127408342718683, 63169771935593153194107

Offset: 0

R. J. Mathar, Mar 07 2022

nonn

a(n) is the number of multigraphs with 2n unlabeled nodes of degree 3 plus 2 noninterchangeable nodes of degree 1, loops allowed. - Andrew Howroyd, Mar 10 2023

R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007, App. D.
R. J. Mathar, Illustrations

Cf. A129427 (no external legs), A352173 (degree 4 case), A361447 (connected).

a(0) prepended and terms a(9) and beyond from Andrew Howroyd, Mar 10 2023

cp's OEIS Frontend

A003175 Almost certainly an erroneous version of A129427.

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A005967 Number of isomorphism classes of connected 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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A129416 Number of isomorphism classes of 3-regular loopless multigraphs of order 2n.

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

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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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