A003400 -id:A003400 - cp's OEIS frontend

A275867 Number of simple disconnected asymmetric graph on n vertices.

0, 0, 0, 0, 0, 0, 8, 144, 3552, 131452, 7840396, 797524408

Offset: 1

Eric W. Weisstein, May 19 2017

nonn

For 2 < n < 12, a(n) = A124059(n-1) (connected asymmetric graphs). This is because the singleton is the only asymmetric graph with fewer than 6 vertices, so in a disconnected asymmetric graph with fewer than 12 vertices one connected component must be the singleton, and it cannot occur more than once. - Falk Hüffner, Jan 16 2020

1

Eric Weisstein's World of Mathematics, Graph Automorphism
Eric Weisstein's World of Mathematics, Identity Graph

Cf. A003400 (not-necessarily connected simple asymmetric graphs).

Cf. A124059 (connected simple asymmetric graphs).

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A003400 = A@003400;
A124059 = A@124059;
a[n_] := A003400[[n]] - A124059[[n]];
a /@ Range[12] (* Jean-François Alcover, Jan 07 2020 *)

a(n) = A003400(n) - A124059(n).

a(12) from Jean-François Alcover, Jan 07 2020

A075094 Triangle of the sorted orders of graph automorphism groups for the simple graphs.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 6, 2, 2, 2, 4, 4, 6, 6, 8, 8, 24, 24, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 12, 12, 12, 12, 12, 12, 24, 24, 120, 120, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Eric W. Weisstein, Aug 31 2002

For n>1 row n ends with n!,n! since the automorphism group of the empty graph and the complete graph is the symmetric group. - Geoffrey Critzer, Aug 09 2016

			From _Geoffrey Critzer_, Aug 09 2016: (Start)
Triangle begins:
  1;
  2, 2;
  2, 2, 6, 6;
  2, 2, 2, 4, 4, 6, 6, 8, 8, 24, 24;
  ... (End)

Sean A. Irvine, Table of n, a(n) for n = 1..13598
Eric Weisstein's World of Mathematics, Graph Automorphism

Cf. A003400, A000088 (row lengths).

a = {1, 2, 4, 11, 34, 156, 1044};
Table[Sort[Table[GraphData[{n, i}, "AutomorphismCount"], {i, 1, a[[n]]}]], {n,1, 7}] // Grid (* Geoffrey Critzer, Aug 09 2016 *)

A352765 Number of asymmetric n-node graphs with the smallest number (A352764(n)) of edges.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 1, 1, 3, 6, 15, 29, 68, 144, 1, 3, 9, 24, 62

Offset: 1

Pontus von Brömssen, Apr 02 2022

Cf. A003400, A352764.

cp's OEIS Frontend

A275867 Number of simple disconnected asymmetric graph on n vertices.

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A075094 Triangle of the sorted orders of graph automorphism groups for the simple graphs.

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A352765 Number of asymmetric n-node graphs with the smallest number (A352764(n)) of edges.

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