A243391 -id:A243391 - cp's OEIS frontend

A334546 Array read by antidiagonals: T(n,k) is the number of unlabeled connected loopless multigraphs with n nodes of degree k or less.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 3, 2, 0, 0, 1, 1, 4, 4, 2, 0, 0, 1, 1, 5, 9, 12, 2, 0, 0, 1, 1, 6, 14, 37, 22, 2, 0, 0, 1, 1, 7, 23, 93, 146, 68, 2, 0, 0, 1, 1, 8, 32, 203, 602, 772, 166, 2, 0, 0, 1, 1, 9, 46, 399, 2126, 5847, 4449, 534, 2, 0, 0

Offset: 0

Andrew Howroyd, May 05 2020

This sequence may be derived from A333893 by inverse Euler transform.

			Array begins:
==============================================
n\k | 0 1 2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1 1   1    1     1      1       1 ...
  1 | 1 1 1   1    1     1      1       1 ...
  2 | 0 1 2   3    4     5      6       7 ...
  3 | 0 0 2   4    9    14     23      32 ...
  4 | 0 0 2  12   37    93    203     399 ...
  5 | 0 0 2  22  146   602   2126    6308 ...
  6 | 0 0 2  68  772  5847  34126  164965 ...
  7 | 0 0 2 166 4449 66289 716141 6021463 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals 0..26)

Columns k=3..5 are A243391, A289157, A334547.
Main diagonal is A334546.
Cf. A289987, A328682 (regular), A333893 (not necessarily connected).

Column k is the inverse Euler transform of column k of A333893.

A289157 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less.

Original entry on oeis.org

1, 1, 4, 9, 37, 146, 772, 4449, 30307, 228605, 1921464, 17652327, 176162548, 1893738334, 21806975279, 267636988052, 3486370839295, 48029272657002, 697542580286159, 10649954607360119, 170508064788069346, 2856122791685125616, 49951625299057923405

Offset: 0

Natan Arie Consigli, Jul 04 2017

nonn

Column k=4 of A334546.
Cf. A121941 (single edges only), A134818 (with no more than triple edges), A289158 (with no more than double edges).
Cf. A243391 (degree 3 or less).

nauty
```
geng -c -D4 ${n} -q | multig -D4 -u
```

a(18)-a(22) from Andrew Howroyd, Mar 20 2020

A243393 Number of isomorphism classes of connected 3-regular loopless simple graphs with n vertices and with semi-edges allowed.

Original entry on oeis.org

1, 1, 2, 6, 10, 29, 64, 194, 531, 1733, 5524, 19430, 69322, 262044, 1016740, 4101318, 16996157, 72556640, 317558689, 1424644848, 6536588420, 30647561117, 146647344812, 715511358833, 3556531372395, 17996244725780, 92634418530686, 484756161038264

Offset: 1

Nico Van Cleemput, Jun 04 2014

nonn

Also: number of isomorphism classes of connected loopless simple graphs with maximum degree at most 3. - Brendan McKay, Mar 11 2020

Nino Bašić and Ivan Damnjanović, On cubic polycirculant nut graphs, arXiv:2411.16904 [math.CO], 2024. See p. 22.
Gunnar Brinkmann, Nico Van Cleemput, and Tomaž Pisanski, Generation of various classes of trivalent graphs, Theoretical Comp. Sci. (2013) Vol. 502, 16-29.

Cf. A121941, A243391, A243392, A243394, A303032.

a(24)-a(28) from Andrew Howroyd, Mar 20 2020

A243392 Number of isomorphism classes of connected 3-regular simple graphs of order 2n with loops and semi-edges allowed.

Original entry on oeis.org

2, 3, 4, 12, 22, 68, 166, 534, 1589, 5464, 18579, 68320, 255424, 1000852, 4018156, 16671976, 70890940, 309439942, 1381815168, 6310880471, 29428287639, 140012980007, 678970863717, 3353545264060, 16857749613964, 86191265140699, 447951112379963, 2365177154077186

Offset: 1

Nico Van Cleemput, Jun 04 2014

nonn

G. Brinkmann, N. Van Cleemput, T. Pisanski, Generation of various classes of trivalent graphs, Theoretical Computer Science 502, 2013, pp.16-29.

Differs from A243391 only in the first term.

a(23)-a(28) from Andrew Howroyd, May 06 2021

A243394 Number of isomorphism classes of connected 3-regular multigraphs with n vertices and with loops and semi-edges allowed.

Original entry on oeis.org

2, 5, 7, 22, 43, 141, 373, 1270, 4053, 14671, 52826, 203289, 795581, 3241367, 13504130, 57904671, 253856990, 1139231977, 5219113084, 24401837085, 116278408069, 564380686932, 2787884851040, 14007277302822, 71538337097031, 371197207327709, 1955833646495247, 10459788214042492

Offset: 1

Nico Van Cleemput, Jun 04 2014

nonn

a(n) is also the number of isomorphism classes of connected multigraphs with n vertices of degree 3 or less and with loops allowed. - Andrew Howroyd, Mar 21 2020

G. Brinkmann, N. Van Cleemput, T. Pisanski, Generation of various classes of trivalent graphs, Theoretical Computer Science 502, 2013, pp.16-29.

Cf. A000421, A005967, A243391, A243392, A243393, A243394.

a(23)-a(28) from Andrew Howroyd, Mar 21 2020

A303030 Number of unlabeled connected loopless multigraphs with n nodes of degree 3 or less and with single or double edges.

Original entry on oeis.org

1, 1, 2, 4, 12, 22, 68, 166, 534, 1589, 5464, 18579, 68320, 255424, 1000852, 4018156, 16671976, 70890940, 309439942, 1381815168, 6310880471, 29428287639, 140012980007, 678970863717, 3353545264060, 16857749613964, 86191265140699, 447951112379963, 2365177154077186

Offset: 0

Natan Arie Consigli, Apr 17 2018

nonn

For n >= 1, a(n) is also the number of hydronitrogen molecules containing only n nitrogen trivalent (octet rule satisfying) atoms. So for example, diazene is counted but hydrazoic acid is not because the former has only trivalent nitrogens and the latter has two non-trivalent nitrogens.
Some of the molecules are theoretical and may or may not exist due to their strained geometries.
Apparently the same as A243391 for n > 2. - Georg Fischer, Oct 16 2018
This is the case since A243391 gives the number of loopless multigraphs with nodes of degree 3 or less. The extra graph in A243391 is the 3-regular graph on 2 nodes. - Andrew Howroyd, Mar 20 2020

			a(3) = 4 because there are 4 molecules satisfying the above condition: triazane, triazene, triazirine, triazidirine.
Note: hydrazoic acid is not counted because there are 2 nitrogens not satisfying the octet rule (one has a positive formal charge and the other one has a negative one).
Graphically, a(3) = 4 because there are 4 graphs satisfying the above condition: the linear graph, the linear graph with one double edge, the triangle graph, and the triangle graph with one double edge. - _Michael B. Porter_, Apr 28 2018

Cf. A134818, A243391, A289158, A303031.

for n in {1..18}; do geng -c -D3 ${n}  -q | multig -m2 -D3 -u;done

a(n) = A243391(n) for n > 2. - Andrew Howroyd, Mar 20 2020

a(20)-a(28) from Andrew Howroyd, Mar 20 2020

A333894 Number of unlabeled loopless multigraphs with n nodes of degree 3 or less.

Original entry on oeis.org

1, 1, 4, 8, 26, 60, 184, 488, 1509, 4468, 14494, 47228, 163584, 580489, 2155219, 8245929, 32708045, 133615420, 562518913, 2432472970, 10794454457, 49060744260, 228113422904, 1083584062308, 5253168117841, 25964917671574, 130732997627332, 669989546176410, 3492408687398352

Offset: 0

Andrew Howroyd, Apr 08 2020

nonn

Column k=3 of A333893.

Cf. A243391.

Euler transform of A243391.

cp's OEIS Frontend

A334546 Array read by antidiagonals: T(n,k) is the number of unlabeled connected loopless multigraphs with n nodes of degree k or less.

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A289157 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less.

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A243393 Number of isomorphism classes of connected 3-regular loopless simple graphs with n vertices and with semi-edges allowed.

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A243392 Number of isomorphism classes of connected 3-regular simple graphs of order 2n with loops and semi-edges allowed.

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A243394 Number of isomorphism classes of connected 3-regular multigraphs with n vertices and with loops and semi-edges allowed.

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Author

Keywords

Comments

Links

Crossrefs

Extensions

A303030 Number of unlabeled connected loopless multigraphs with n nodes of degree 3 or less and with single or double edges.

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Programs

nauty

Formula

Extensions

A333894 Number of unlabeled loopless multigraphs with n nodes of degree 3 or less.

Views

Author

Keywords

Crossrefs

Formula