A129432 -id:A129432 - cp's OEIS frontend

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

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

A085549 Number of isomorphism classes of connected 4-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 2, 4, 10, 28, 97, 359, 1635, 8296, 48432, 316520, 2305104, 18428254, 160384348, 1506613063, 15180782537, 163211097958, 1864251304892, 22540603640086, 287577260214946, 3860595341568062, 54397355465967057, 802684717378090204

Offset: 1

Benjamin A. Burton (bab(AT)debian.org), Jul 04 2003

Also the number of different potential face pairing graphs for closed 3-manifold triangulations.

Computed from A129429 by an inverse Euler transform. - R. J. Mathar, Mar 09 2019

B. A. Burton, Minimal triangulations and face pairing graphs, preprint, 2003.

Andrew Howroyd, Table of n, a(n) for n = 1..40
B. A. Burton, Regina (3-manifold topology software).
B. A. Burton, Minimal triangulations and normal surfaces, Ph.D. thesis, University of Melbourne, 2003.
B. A. Burton, Face pairing graphs and 3-manifold enumeration, arXiv:math/0307382 [math.GT], 2003.
B. A. Burton, Enumeration of non-orientable 3-manifolds using face-pairing graphs and union-find, Discrete and Computational Geometry, 38 (2007), 527-571.
R. de Mello Koch, S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007
H. Kleinert, A. Pelster, B. Kastening, M. Bachmann, Recursive graphical construction of Feynman diagrams and their multiplicities in Phi^4 and Phi^2*A theory, Phys. Rev. E 62 (2) (2000), 1537 eq (4.20) or arXiv:hep-th/9907168, 1999.
B. Martelli and C. Petronio, Three-manifolds having complexity at most 9, Experiment. Math., Vol. 10 (2001), pp. 207-236
R. J. Mathar, Illustrations

Column k=4 of A333397.

Cf. A129429, A129417, A005967, A129430, A129432, A129434, A129436, A118560.

A129429 = Cases[Import["https://oeis.org/A129429/b129429.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[A129429] (* Jean-François Alcover, Dec 03 2019, updated Mar 17 2020 *)

Inverse Euler transform of A129429.

a(12)-a(16) from Brendan McKay, Apr 15 2007, computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/
Edited by N. J. A. Sloane, Oct 01 2007
a(17)-a(23) from A129429 from Jean-François Alcover, Dec 03 2019

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

A129430 Number of isomorphism classes of connected 5-regular multigraphs of order 2n, loops allowed.

Original entry on oeis.org

3, 26, 639, 40264, 5846105, 1620621150, 752480161278, 538934691750368, 562620407713724992, 820458681175954269942, 1616087981640640784235446, 4183688192689449962777539596, 13914233045360143936837907106395, 58319096569220501055727735345999221

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 A333397.

Cf. A129431, A129419, A005967, A085549, A129432, A129434, A129436.

Inverse Euler transform of A129431. - Andrew Howroyd, Mar 19 2020

a(8)-a(14) added by Andrew Howroyd, Mar 21 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.

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

A129434 Number of isomorphism classes of connected 7-regular multigraphs of order 2n, loops allowed.

Original entry on oeis.org

4, 91, 12398, 8530044, 20068725095, 122563246940846, 1657847267734501346, 44557979504639651662163, 2193071655191529316254072193, 185380797361862371952777763438426

Offset: 1

Brendan McKay, Apr 15 2007

nonn

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

Column k=7 of A333397.

Cf. A129435, A129423, A005967, A085549, A129430, A129432, A129436.

Inverse Euler transform of A129435. - Andrew Howroyd, Mar 19 2020

a(6)-a(10) added by 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

cp's OEIS Frontend

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

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A085549 Number of isomorphism classes of connected 4-regular multigraphs of order n, loops allowed.

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

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nauty

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A129430 Number of isomorphism classes of connected 5-regular multigraphs of order 2n, loops allowed.

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

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A129434 Number of isomorphism classes of connected 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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