A124059 -id:A124059 - cp's OEIS frontend

A241464 Number of simple connected graphs g on n nodes with |Aut(g)| = 36.

0, 0, 0, 0, 0, 1, 3, 16, 132, 1341

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A241465 Number of simple connected graphs g on n nodes with |Aut(g)| = 48.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 14, 65, 504, 5215

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A241466 Number of simple connected graphs g on n nodes with |Aut(g)| = 72.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 16, 124, 1070

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A241467 Number of simple connected graphs g on n nodes with |Aut(g)| = 120.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 5, 21, 211

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A241468 Number of simple connected graphs g on n nodes with |Aut(g)| = 144.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 12, 51, 477

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A241469 Number of simple connected graphs g on n nodes with |Aut(g)| = 240.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 8, 51, 336

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A241470 Number of simple connected graphs g on n nodes with |Aut(g)| = 720.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 13, 60

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A241471 Number of simple connected graphs g on n nodes with |Aut(g)| = 5040.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 5

Offset: 1

Travis Hoppe and Anna Petrone, Apr 25 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Graph Automorphism

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.

A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.

Original entry on oeis.org

1, 1, 2, 6, 17, 52, 173, 585, 2064, 7520, 28265, 109501, 437394, 1799843, 7629463, 33302834, 149633151, 691702799, 3287804961, 16058229900, 80533510224, 414384339438, 2185878202630, 11811050484851, 65318772618624, 369428031895444, 2135166786135671, 12601624505404858

Offset: 0

Gus Wiseman, Jan 05 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}        {{1},{2},{3},{4}}
             {{1},{1,2}}  {{1},{2},{1,2}}      {{1},{2},{3},{1,2}}
                          {{1},{2},{1,3}}      {{1},{2},{3},{1,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}    {{1},{2},{1,2},{3,4}}
                          {{1,2},{1,3},{2,3}}  {{1},{2},{1,3},{1,4}}
                                               {{1},{2},{1,3},{2,3}}
                                               {{1},{2},{1,3},{2,4}}
                                               {{1},{3},{1,2},{2,4}}
                                               {{1},{1,2},{1,3},{1,4}}
                                               {{1},{1,2},{1,3},{2,3}}
                                               {{1},{1,2},{1,3},{2,4}}
                                               {{1},{1,2},{2,3},{3,4}}
                                               {{2},{1,2},{1,3},{1,4}}
                                               {{4},{1,2},{1,3},{2,3}}
                                               {{1,2},{1,3},{1,4},{2,3}}
                                               {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Loop.

For any number of edges of any size we have A000612, covering A055621.
For any number of edges we have A000666, A054921, A322700.
The labeled version is A014068.
Counting by weight gives A320663, or A339888 with loops {x,x}.
The covering case is A368599.
For edges of any size we have A368731, covering A368186.
Row sums of A368836.
A000085 counts set partitions into singletons or pairs.
A001515 counts length-n set partitions into singletons or pairs.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A001434, A007716, A007717, A058891, A070166, A122848, A124059, A283877, A302545, A322661, A339741, A339887, A370168.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,2}],{n}]]],{n,0,5}]

a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024

a(n) = A070166(n, n). - Andrew Howroyd, Jan 09 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 09 2024

A003400 Number of asymmetric (not necessarily connected) graphs with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 8, 152, 3696, 135004, 7971848, 805364776, 144123121972

Offset: 1

N. J. A. Sloane

Number of simple graphs g on n nodes with |Aut(g)| = 1.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 220, Section P3.4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Klaus Brockhaus, The 6-node asymmetric graphs
Zoran Maksimovic, Number of graphs on n nodes whose automorphism group orders are k, n<=11
Yoav Spector, Moshe Schwartz, Study of potential Hamiltonians for quantum graphity, arXiv:1808.05632 [cond-mat.stat-mech], 2018.
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, Graph Automorphism
Eric Weisstein's World of Mathematics, Identity Graph

Cf. A124059 (connected simple asymmetric graphs).
Cf. A275867 (disconnected simple asymmetric graphs).
Cf. A000088 (simple graphs).

for n in {1..10}; do geng -q ${n} | countg -q -a1 | grep altogether | awk '{print $1}'; done # - Sean A. Irvine, Apr 22 2015

a(n) = A124059(n) + A275867(n).

a(8) and a(9) from Eric W. Weisstein, Jun 09 2004
a(10) and a(11) from Zoran Maksimovic, Vladeta Jovovic, Jan 21 2005
a(12) from Sean A. Irvine, Apr 22 2015

cp's OEIS Frontend

A241464 Number of simple connected graphs g on n nodes with |Aut(g)| = 36.

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A241465 Number of simple connected graphs g on n nodes with |Aut(g)| = 48.

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A241466 Number of simple connected graphs g on n nodes with |Aut(g)| = 72.

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A241467 Number of simple connected graphs g on n nodes with |Aut(g)| = 120.

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A241468 Number of simple connected graphs g on n nodes with |Aut(g)| = 144.

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A241469 Number of simple connected graphs g on n nodes with |Aut(g)| = 240.

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A241470 Number of simple connected graphs g on n nodes with |Aut(g)| = 720.

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A241471 Number of simple connected graphs g on n nodes with |Aut(g)| = 5040.

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A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.

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A003400 Number of asymmetric (not necessarily connected) graphs with n nodes.

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nauty

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