A289157 -id:A289157 - cp's OEIS frontend

A121941 Number of unlabeled connected simple graphs with n nodes of degree 4 or less.

1, 1, 1, 2, 6, 21, 78, 353, 1929, 12207, 89402, 739335, 6800637, 68531618, 748592936, 8788983173, 110201690911, 1468157196474, 20695559603921, 307590282700915, 4805537369573319, 78710267083015571, 1348394635886684901, 24109112440149231355, 449050443283294835914

Offset: 0

Parthasarathy Nambi, Sep 03 2006

Number of graphs of hydrogen bonded water clusters.
This counts the connected graphs where each vertex has degree 4 or less. - Charles R Greathouse IV, Jul 07 2017
Also number of saturated hydrocarbons and allotropes with n carbon, or valence 4, atoms (excluding stereoisomers) following the octet rule. - Natan Arie Consigli, Jul 07 2017

			With 4 carbons, n-butane, i-butane, cyclobutane, bicyclobutane, methylcyclopropane and tetrahedrane are the 6 isomers satisfying the property above, so a(4)=6. - _Natan Arie Consigli_, Jul 07 2017
If n=5 then the number of graphs of hydrogen bonded water clusters is 21.

G. Brinkmann, Generating water clusters and other directed graphs, J. Math. Chem. 46 (4) (2009) 1112-1121, Table 1.
T. Miyake and M. Aida, Enumeration of topology-distinct structures of hydrogen bonded water clusters, Chem. Phys. Lett., vol. 363 (2002) pp. 106-110. See Table 1 column 2 on page 109.

Cf. A121942, A243393 (degree 3 or less), A287424 (excludes allotropes), A289157, A289158, A303032 (degree 5 or less).

nauty
```
geng -c -D4 ${n} -q | multig -m1 -D4 -u
```

More terms sent by Natan Arie Consigli, Jul 07 2017
Renamed by Andrew Howroyd, Mar 19 2020 based on comment by Charles R Greathouse IV.
a(16)-a(24) from Andrew Howroyd, Mar 19 2020

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

Original entry on oeis.org

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.

A134818 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less and with at most triple edges.

Original entry on oeis.org

1, 3, 9, 37, 146, 772, 4449, 30307, 228605, 1921464, 17652327, 176162548, 1893738334, 21806975279, 267636988052, 3486370839295, 48029272657002, 697542580286159, 10649954607360119, 170508064788069346, 2856122791685125616, 49951625299057923405

Offset: 1

David Consiglio, Jr., Jan 28 2008

From Natan Arie Consigli, May 29 2017: (Start)
Original name was "Number of hydrocarbon structures that can be drawn (excluding stereoisomers)" but this has been replaced with a mathematical definition which is more consistent with the terms of the sequence and the program.
In chemical terms this counts the following, given n carbon atoms:
- carbon allotropes;
- aliphatic hydrocarbons;
- resonance structures of graphically non-equivalent anti-aromatic and aromatic hydrocarbons.
Some molecules are theoretical and may or may not exist.
(End)
Computed over a period of several years and confirmed using the Molgen program.
Terms for n = 8,9,10 calculated using an exhaustive algorithm and Nauty. The algorithm correctly found the 7 known terms and the known acyclic hydrocarbons (up to n=10, see A002986) were extracted from the results correctly. - Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 17 2008
Except for a(2), the same as A289157. The extra graph in A289157 is the 4-regular graph on 2 nodes. - Andrew Howroyd, Mar 20 2020

			For n = 2 there are a(2) = 3 structures that can be drawn with 2 carbons (ethane, ethene, and ethyne).
For n = 7 there are a(7) = 4449 structures that can be drawn with 7 carbons.

Brendan McKay, Nauty
Molgen, Publications

Cf. A134819 gives the number of possible structures, broken down by units of unsaturation.
Cf. A002986 (non-cyclic hydrocarbons).
Cf. A121941, A289157, A289158, A303033.

nauty
```
geng -c -D4 ${n} -q | multig -m3 -D4 -u
```

a(n) = A289157(n) for n > 2. - Andrew Howroyd, Mar 20 2020

a(8)-a(10) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 17 2008
a(11) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 24 2008
a(12) sent by David Consiglio, Jr., Apr 23 2008
a(12) corrected, a(13) and a(14) added - David Consiglio, Jr. Nov 03 2011
a(15)-a(17) computed using nauty by Sean A. Irvine, Jan 19 2015
New name from Natan Arie Consigli, May 29 2016
a(18)-a(22) from Andrew Howroyd, Mar 20 2020

A289158 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less and with at most double edges.

Original entry on oeis.org

1, 1, 2, 7, 28, 112, 590, 3419, 23453, 178599, 1516692, 14083855, 142029043, 1542152723, 17925912574, 221938298129, 2914638247016, 40455853460661, 591654481313077, 9091698010380468, 146433114305147508, 2466517505722469501, 43361349681960337334

Offset: 0

Natan Arie Consigli, Jul 04 2017

nonn

In chemical terms this counts the following molecules (excluding stereoisomers) without triple bonds, given n carbon atoms:
- carbon allotropes;
- aliphatic hydrocarbons;
- resonance structures of graphically non-equivalent anti-aromatic and aromatic hydrocarbons;
Some molecules are theoretical and may or may not exist.

Daniel R. Herber, Enhancements to the perfect matching approach for graph enumeration-based engineering challenges, Proceedings of the ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE 2020).

Cf. A121941, A289157 (allows more than two edges), A303030, A303031.

nauty
```
geng -c -D4 ${n} -q | multig -m2 -D4 -u
```

a(15)-a(22) from Andrew Howroyd, Mar 20 2020

A289988 Number of unlabeled connected loopless multigraphs with n nodes of degree n or less.

Original entry on oeis.org

1, 1, 2, 4, 37, 602, 34126, 6021463, 3616906549, 7361925161868, 51324462383008758, 1240420936122453106498, 105141919479926837860474091, 31581183353539008502807807352728

Offset: 0

Natan Arie Consigli, Aug 19 2017

Multigraphs are loopless.

Main diagonal of A334546.

Cf. A007719, A076864, A134818, A289157, A289158, A289987.

for n in {1..8}; do geng -c -D${n} ${n} -q | multig -m$[${n}-1] -D$[${n}-1] -u; done

a(0) corrected and a(9)-a(13) from Andrew Howroyd, May 05 2020

A333895 Number of unlabeled loopless multigraphs with n nodes of degree 4 or less.

Original entry on oeis.org

1, 1, 5, 14, 61, 243, 1228, 6684, 42681, 304569, 2445296, 21676143, 210409238, 2213794888, 25064493886, 303468325469, 3909518220091, 53364222428953, 769009695515272, 11663042823897054, 185651491398956596, 3094082786005252494, 53871873427160960819

Offset: 0

Andrew Howroyd, Apr 08 2020

nonn

Column k=4 of A333893.

Cf. A289157.

Euler transform of A289157.

cp's OEIS Frontend

A121941 Number of unlabeled connected simple graphs with n nodes of degree 4 or less.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

nauty

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A134818 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less and with at most triple edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

nauty

Formula

Extensions

A289158 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less and with at most double edges.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

nauty

Extensions

A289988 Number of unlabeled connected loopless multigraphs with n nodes of degree n or less.

Views

Author

Keywords

Comments

Crossrefs

Programs

nauty

Extensions

A333895 Number of unlabeled loopless multigraphs with n nodes of degree 4 or less.

Views

Author

Keywords

Crossrefs

Formula