A000421 -id:A000421 - cp's OEIS frontend

A005965 Erroneous version of A000421.

Original entry on oeis.org

1, 2, 6, 20, 91, 506

Offset: 1

N. J. A. Sloane

dead

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).

A210690 Erroneous version of A000421.

Original entry on oeis.org

1, 2, 6, 20, 93

Offset: 1

N. J. A. Sloane, Mar 30 2012

dead

Cadogan, C. C. Graph-lattices and the enumeration of linear graphs. 1970 Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1970) pp. 61--86 Louisiana State Univ., Baton Rouge, La. MR0269553 (42 #4448).

A002851 Number of unlabeled trivalent (or cubic) connected simple graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 5, 19, 85, 509, 4060, 41301, 510489, 7319447, 117940535, 2094480864, 40497138011, 845480228069, 18941522184590, 453090162062723, 11523392072541432, 310467244165539782, 8832736318937756165

Offset: 0

N. J. A. Sloane

			G.f. = 1 + x^2 + 2*x^3 + 5*x^4 + 19*x^5 + 85*x^6 + 509*x^7 + 4060*x^8 + 41302*x^9 + 510489*x^10 + 7319447*x^11 + ...
a(0) = 1 because the null graph (with no vertices) is vacuously 3-regular.
a(1) = 0 because there are no simple connected cubic graphs with 2 nodes.
a(2) = 1 because the tetrahedron is the only cubic graph with 4 nodes.
a(3) = 2 because there are two simple cubic graphs with 6 nodes: the bipartite graph K_{3,3} and the triangular prism graph.

CRC Handbook of Combinatorial Designs, 1996, p. 647.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 195.
R. C. Read, Some applications of computers in graph theory, in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, pp. 417-444.
R. C. Read and G. F. Royle, Chromatic roots of families of graphs, pp. 1009-1029 of Y. Alavi et al., eds., Graph Theory, Combinatorics and Applications. Wiley, NY, 2 vols., 1991.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence)

Peter Adams, Ryan C. Bunge, Roger B. Eggleton, Saad I. El-Zanati, Uğur Odabaşi, and Wannasiri Wannasit, Decompositions of complete graphs and complete bipartite graphs into bipartite cubic graphs of order at most 12, Bull. Inst. Combinatorics and Applications (2021) Vol. 92, 50-61.
G. Brinkmann, J. Goedgebeur and B. D. McKay, Generation of cubic graphs, Discr. Math. Theor. Comp. Sci. 13 (2) (2011) 69-80
G. Brinkmann, J. Goedgebeur, and N. Van Cleemput, The history of the generation of cubic graphs, Int. J. Chem. Modeling 5 (2-3) (2013) 67-89
F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
F. C. Bussemaker, S. Cobeljic, D. M. Cvetkovic, and J. J. Seidel, Cubic graphs on <= 14 vertices J. Combinatorial Theory Ser. B 23(1977), no. 2-3, 234--235. MR0485524 (58 #5354).
Timothy B. P. Clark and Adrian Del Maestro, Moments of the inverse participation ratio for the Laplacian on finite regular graphs, arXiv:1506.02048 [math-ph], 2015.
Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 1.
H. Gropp, Enumeration of regular graphs 100 years ago, Discrete Math., 101 (1992), 73-85.
House of Graphs, Cubic graphs
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Klin, M. Rücker, Ch. Rücker and G. Tinhofer, Algebraic Combinatorics [broken link]
M. Klin, M. Rücker, Ch. Rücker, and G. Tinhofer, Algebraic Combinatorics (1997)
Denis S. Krotov and Konstantin V. Vorob'ev, On unbalanced Boolean functions attaining the bound 2n/3-1 on the correlation immunity, arXiv:1812.02166 [math.CO], 2018.
R. J. Mathar/Wikipedia, Table of simple cubic graphs [From _N. J. A. Sloane_, Feb 28 2012]
M. Meringer, Tables of Regular Graphs
R. W. Robinson and N. C. Wormald, Numbers of cubic graphs. J. Graph Theory 7 (1983), no. 4, 463-467.
Sage, Common Graphs (Graph Generators)
J. J. Seidel, R. R. Korfhage, & N. J. A. Sloane, Correspondence 1975
H. M. Sultan, Separating pants decompositions in the pants complex.
H. M. Sultan, Net of Pants Decompositions Containing a non-trivial Separating Curve in the Pants Complex, arXiv:1106.1472 [math.GT], 2011.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Cubic Graph

Cf. A004109 (labeled connected cubic), A361407 (rooted connected cubic), A321305 (signed connected cubic), A000421 (connected cubic loopless multigraphs), A005967 (connected cubic multigraphs), A275744 (multisets).
Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
3-regular simple graphs: this sequence (connected), A165653 (disconnected), A005638 (not necessarily connected), A005964 (planar).
Connected regular graphs A005177 (any degree), A068934 (triangular array), specified degree k: this sequence (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: this sequence (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7). (End)

More terms from Ronald C. Read

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 6, 21, 94, 540, 4207, 42110, 516344, 7373924, 118573592, 2103205738, 40634185402, 847871397424, 18987149095005, 454032821688754, 11544329612485981, 310964453836198311, 8845303172513781271

Offset: 0

N. J. A. Sloane

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-4)-regular graphs on 2n vertices.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Jason Kimberley, Not-necessarily connected regular graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
R. W. Robinson, Cubic graphs (notes)
Robinson, R. W.; Wormald, N. C., Numbers of cubic graphs, J. Graph Theory 7 (1983), no. 4, 463-467.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Cubic Graph
Gal Weitz, Lirandë Pira, Chris Ferrie, and Joshua Combes, Sub-universal variational circuits for combinatorial optimization problems, arXiv:2308.14981 [quant-ph], 2023.

Cf. A000421.
Row sums of A275744.
3-regular simple graphs: A002851 (connected), A165653 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), this sequence (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Not necessarily connected 3-regular simple graphs with girth *at least* g: this sequence (g=3), A185334 (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A002851(n) + A165653(n).

This sequence is the Euler transformation of A002851.

More terms from Ronald C. Read.

Comment, formulas, and (most) crossrefs by Jason Kimberley, 2009 and 2012

A328682 Array read by antidiagonals: T(n,r) is the number of connected r-regular loopless multigraphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 0, 1, 0, 1, 1, 4, 6, 6, 1, 0, 0, 1, 0, 1, 0, 6, 0, 19, 0, 1, 0, 0, 1, 0, 1, 1, 7, 15, 49, 50, 20, 1, 0, 0, 1, 0, 1, 0, 9, 0, 120, 0, 204, 0, 1, 0, 0, 1, 0, 1, 1, 11, 36, 263, 933, 1689, 832, 91, 1, 0, 0, 1, 0, 1, 0, 13, 0, 571, 0, 13303, 0, 4330, 0, 1, 0, 0, 1, 0, 1, 1, 15, 72, 1149, 12465, 90614, 252207, 187392, 25227, 509, 1, 0, 0

Offset: 0

Natan Arie Consigli, Dec 17 2019

Initial terms computed using 'Nauty and Traces' (see the link).
T(0,r) = 1 because the "nodeless" graph has zero (therefore in this case all) nodes of degree r (for any r).
T(1,0) = 1 because only the empty graph on one node is 0-regular on 1 node.
T(1,r) = 0, for r>0: there's only one node and loops aren't allowed.
T(2,r) = 1, for r>0 since the only edges that are allowed are between the only two nodes.
T(3,r) = parity of r, for r>0. There are no such graphs of odd degree and for an even degree the only multigraph satisfying that condition is the regular triangular multigraph.
T(n,0) = 0, for n>1 because graphs having more than a node of degree zero are disconnected.
T(n,1) = 0, for n>2 since any connected graph with more than two nodes must have a node of degree greater than two.
T(n,2) = 1, for n>1: the only graphs satisfying that condition are the cyclic graphs of order n.
This sequence may be derived from A333330 by inverse Euler transform. - Andrew Howroyd, Mar 15 2020

			Square matrix T(n,r) begins:
========================================================
n\r | 0     1     2     3     4     5      6      7
----+---------------------------------------------------
  0 | 1,    1,    1,    1,    1,    1,     1,     1, ...
  1 | 1,    0,    0,    0,    0,    0,     0,     0, ...
  2 | 0,    1,    1,    1,    1,    1,     1,     1, ...
  3 | 0,    0,    1,    0,    1,    0,     1,     0, ...
  4 | 0,    0,    1,    2,    3,    4,     6,     7, ...
  5 | 0,    0,    1,    0,    6,    0,    15,     0, ...
  6 | 0,    0,    1,    6,   19,   49,   120,   263, ...
  7 | 0,    0,    1,    0,   50,    0,   933,     0, ...
  8 | 0,    0,    1,   20,  204, 1689, 13303, 90614, ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..350
Brendan McKay and Adolfo Piperno, Nauty and Traces

Rows n=3..9 are: A000035, A253186, A324221, A324218, A324217, A325474, A327604.
Columns r=3..8 are: A000421, A129417, A129419, A129421, A129423, A129425.
Cf. A289986 (main diagonal), A333330 (not necessarily connected), A333397.

# This program will execute the "else echo" line if the graph is nontrivial (first three columns, first two rows or both row and column indices are odd)
for ((i=0; i<16; i++)); do
n=0
r=${i}
while ((n<=i)); do
if( (((r==0)) && ((n==0)) ) || ( ((r==0)) && ((n==1)) ) || ( ((r==1)) && ((n==2)) ) || ( ((r==2)) && !((n==1)) ) ); then
echo 1
elif( ((n==0)) || ((n==1)) || ((r==0)) || ((r==1)) || (! ((${r}%2 == 0)) && ! ((${n}%2 == 0)) || ( ((r==2)) && ((n==1)) )) ); then
echo 0
else echo $(./geng -c -d1 ${n} -q | ./multig -m${r} -r${r} -u 2>&1 | cut -d ' ' -f 7 | grep -v '^$');  fi;
((n++))
((r--))
done
done

Column r is the inverse Euler transform of column r of A333330. - Andrew Howroyd, Mar 15 2020

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

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

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

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

cp's OEIS Frontend

A005965 Erroneous version of A000421.

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A210690 Erroneous version of A000421.

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A002851 Number of unlabeled trivalent (or cubic) connected simple graphs with 2n nodes.

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A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

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A328682 Array read by antidiagonals: T(n,r) is the number of connected r-regular loopless multigraphs on n unlabeled nodes.

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

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

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

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

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nauty

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