cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A124059 Number of connected asymmetric graphs with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 8, 144, 3552, 131452, 7840396, 797524380, 143325597564
Offset: 1

Views

Author

Franklin T. Adams-Watters, Nov 03 2006

Keywords

Comments

A graph possessing only a single automorphism is called an identity or asymmetric graph, |Aut(g)|=1. - Travis Hoppe, Apr 27 2014

Links

Crossrefs

Cf. A003400 (not-necessarily connected simple asymmetric graphs).
Cf. A275867 (disconnected simple asymmetric graphs).
Cf. Values of |Aut(g)| for simple connected graphs, A124059, A241454, A241455, A241456, A241457, A241458, A241459, A241460, A241461, A241462, A241463, A241464, A241465, A241466, A241467, A241468, A241469, A241470, A241471.

Formula

Inverse WEIGH transform of A003400 (see Transforms link).
a(n) = A003400(n) - A275867(n).

Extensions

a(12) from Alois P. Heinz, Jun 11 2018

A075095 Number of simple graphs g on n nodes with |Aut(g)| = 2.

Original entry on oeis.org

0, 2, 2, 3, 11, 46, 354, 4431, 89004, 3026014, 179282248, 18885632962
Offset: 1

Views

Author

Eric W. Weisstein, Aug 31 2002

Keywords

Links

Crossrefs

Cf. A003400, A075096, A075097, A075098.

Extensions

a(8)-a(9) from Eric W. Weisstein, Jun 09 2004
a(10)-a(11) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 04 2008
a(12) from Sean A. Irvine, Feb 09 2025

A330297 Number of labeled simple graphs covering n vertices with exactly two automorphisms, or with exactly n!/2 graphs obtainable by permuting the vertices.

Original entry on oeis.org

0, 0, 1, 3, 24, 540, 13320
Offset: 0

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Comments

These are graphs with exactly one involution and no other symmetries.

Examples

			The a(4) = 24 graphs:
  {12,13,24}  {12,13,14,23}
  {12,13,34}  {12,13,14,24}
  {12,14,23}  {12,13,14,34}
  {12,14,34}  {12,13,23,24}
  {12,23,34}  {12,13,23,34}
  {12,24,34}  {12,14,23,24}
  {13,14,23}  {12,14,24,34}
  {13,14,24}  {12,23,24,34}
  {13,23,24}  {13,14,23,34}
  {13,24,34}  {13,14,24,34}
  {14,23,24}  {13,23,24,34}
  {14,23,34}  {14,23,24,34}

Links

Crossrefs

The non-covering version is A330345.
The unlabeled version is A330346 (not A241454).
Covering simple graphs are A006129.
Covering graphs with exactly one automorphism are A330343.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), and A330346 (unlabeled covering).
Cf. A003400, A006125, A016031, A124059, A143543, A241454, A330098, A330229, A330230, A330231, A330233.

Programs

  • Mathematica
    graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[graprms[#]]==n!/2&]],{n,0,5}]

Formula

a(n) = n!/2 * A330346(n).

A330344 Number of unlabeled graphs with n vertices whose covered portion has exactly two automorphisms.

Original entry on oeis.org

0, 1, 2, 4, 13, 50, 367
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 graphs:
  {12}  {12}     {12}           {12}
        {12,13}  {12,13}        {12,13}
                 {12,13,24}     {12,13,24}
                 {12,13,14,23}  {12,13,14,23}
                                {12,13,14,25}
                                {12,13,24,35}
                                {12,13,14,23,25}
                                {12,13,14,23,45}
                                {12,13,15,24,34}
                                {12,13,14,15,23,24}
                                {12,13,14,23,24,35}
                                {12,13,14,23,25,45}
                                {12,13,14,15,23,24,35}

Crossrefs

The labeled version is A330345.
The covering case is A330346 (not A241454).
Unlabeled graphs are A000088.
Unlabeled graphs with exactly one automorphism are A003400.
Unlabeled connected graphs with exactly one automorphism are A124059.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), and A330346 (unlabeled covering).
Cf. A000612, A004111, A055621, A241454, A283877, A330098, A330227, A330230, A330231, A330233, A330294, A330295.

Formula

Partial sums of A330346.

A330346 Number of unlabeled simple graphs covering n vertices with exactly two automorphisms.

Original entry on oeis.org

0, 0, 1, 1, 2, 9, 37
Offset: 0

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Comments

First differs from A241454 at n = 8.

Examples

			Non-isomorphic representatives of the a(5) = 9 graphs:
  {12,13,14,25}
  {12,13,24,35}
  {12,13,14,23,25}
  {12,13,14,23,45}
  {12,13,15,24,34}
  {12,13,14,15,23,24}
  {12,13,14,23,24,35}
  {12,13,14,23,25,45}
  {12,13,14,15,23,24,35}

Links

Crossrefs

The labeled version is A330297.
The non-covering version is A330344.
Unlabeled covering graphs are A002494.
Unlabeled connected graphs with exactly two automorphisms are A241454.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).
Cf. A000088, A003400, A055621, A124059, A330098, A330227, A330230, A330231, A330294, A330295, A330343.

A051504 Number of asymmetric digraphs with n nodes.

Original entry on oeis.org

1, 1, 1, 7, 136, 8001, 1445297, 863488287
Offset: 0

Views

Author

Vladeta Jovovic

Keywords

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 231.
  • P. K. Stockmeyer, The enumeration of graphs with prescribed automorphism group, Dissertation, Univ. of Michigan, Ann Arbor, 1971.

Crossrefs

Cf. A000273 (digraphs), A003400 (asymmetric graphs), A030242 (asymmetric relations), A067309.

Formula

a(n) = A000273(n) - A067309(n). - Andrew Howroyd, Dec 06 2020

Extensions

a(6) from Christian G. Bower Dec 15 1999
a(7) from Andrew Howroyd, Dec 06 2020

A075096 Number of simple graphs g on n nodes with |Aut(g)| = 4.

Original entry on oeis.org

0, 0, 0, 2, 6, 36, 248, 2264, 31754, 710574, 26501530, 1712953820
Offset: 1

Views

Author

Eric W. Weisstein, Aug 31 2002

Keywords

Links

Crossrefs

Cf. A003400, A075095, A075097, A075098.

Extensions

a(8)-a(9) from Eric W. Weisstein, Jun 09 2004
a(10)-a(11) from Maksimovic's preprint added by Max Alekseyev, Apr 26 2010
a(12) from Sean A. Irvine, Feb 09 2025

A075097 Number of simple graphs g on n nodes with |Aut(g)| = 6.

Original entry on oeis.org

0, 0, 2, 2, 2, 8, 38, 252, 3262, 71362, 2626418, 164764114
Offset: 1

Views

Author

Eric W. Weisstein, Aug 31 2002

Keywords

Links

Crossrefs

Cf. A003400, A075095, A075096, A075098.

Extensions

a(8)-a(9) from Eric W. Weisstein, Jun 09 2004
a(10)-a(11) from Maksimovic's preprint added by Max Alekseyev, Apr 26 2010
a(12) from Sean A. Irvine, Feb 10 2025

A075098 Number of simple graphs g on n nodes with |Aut(g)| = 8.

Original entry on oeis.org

0, 0, 0, 2, 4, 14, 74, 623, 7003, 117914, 3079400, 131895854
Offset: 1

Views

Author

Eric W. Weisstein, Aug 31 2002

Keywords

Links

Crossrefs

Cf. A003400, A075095, A075096, A075097.

Extensions

a(8)-(9) from Eric W. Weisstein, Jun 09 2004
a(10)-a(11) from Maksimovic's preprint added by Max Alekseyev, Apr 26 2010
a(12) from Sean A. Irvine, Feb 10 2025

A330343 Number of labeled fully chiral simple graphs (also called identity or asymmetric graphs) covering n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 5760, 766080, 149022720, 48990251520, 28928242022400, 32147584690636800, 69035206021583155200
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Comments

In a fully chiral graph, every permutation of the vertices gives a different representative, so the only automorphism is the identity.

Crossrefs

The unlabeled version is A003400.
Identity trees are A004111.
Covering simple graphs are A006129.
Full chiral integer partitions are A330228.
Fully chiral factorizations are A330235.
Fully chiral set-systems are A330229 (labeled covering), A330282 (labeled), A330294 (unlabeled), A330295 (unlabeled covering).
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).
Cf. A006125, A016031, A124059, A143543, A330098, A330224, A330226, A330227, A330230, A330231, A330236.

Programs

  • Mathematica
    graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[graprms[#]]==n!&]],{n,5}] (* brute force, not for computation *)

Formula

a(n) = n! * A003400(n).
Showing 1-10 of 13 results. Next

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