A129421 -id:A129421 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

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

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

A129422 Number of isomorphism classes of 6-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 7, 16, 128, 955, 13467, 253373, 6466074, 205749149, 7943313377, 363853255012, 19485170158346, 1205488841884007, 85308028236495340, 6846462326434510551, 618498122199399056707, 62478078728492272712838, 7015617595855429187696753

Offset: 1

Brendan McKay, Apr 15 2007

nonn

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

Column k=6 of A333330.

Cf. A129421, A129433, A129416, A129418, A129420, A129424, A129426.

Euler transform of A129421. - Andrew Howroyd, Mar 14 2020

a(1)=0 prepended and a(14)-a(20) from Andrew Howroyd, Mar 17 2020

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

A129432 Number of isomorphism classes of connected 6-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 3, 9, 47, 291, 2789, 35646, 622457, 14019433, 395208047, 13561118011, 555498075986, 26751985389463, 1496090275853092, 96154662330195078, 7038800665162854369, 582281978355495520076, 54057819690711609171892, 5597375885970846586170796, 642829784413912305507730345

Offset: 1

Brendan McKay, Apr 15 2007

nonn

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

Column k=6 of A333397.

Cf. A129433, A129421, A005967, A085549, A129430, A129434, A129436.

Inverse Euler transform of A129433. - Andrew Howroyd, Mar 19 2020

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

A289986 Number of connected 2n-regular loopless multigraphs on 2n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 120, 543116, 635669057538, 112368754788708539549

Offset: 0

Natan Arie Consigli, Aug 19 2017

Multigraphs are loopless.

There are no (2n+1)-regular multigraphs with (2n+1) number of points, for every nonnegative n.

Cf. A005177, A129417, A129421, A129425, A328682.

for n in {1..4}; do geng -c -d1 $[2*$n] -q | multig -m$[2*$n] -r$[2*$n] -u; done

a(n) = A328682(2*n, 2*n). - Andrew Howroyd, Mar 18 2020

a(5)-a(6) from Andrew Howroyd, Mar 18 2020

cp's OEIS Frontend

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

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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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A129425 Number of isomorphism classes of connected 8-regular loopless multigraphs of order n.

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A129422 Number of isomorphism classes of 6-regular loopless multigraphs of order n.

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

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nauty

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A129432 Number of isomorphism classes of connected 6-regular multigraphs of order n, loops allowed.

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