A327367 Number of labeled simple graphs with n vertices, at least one of which is isolated.

0, 1, 1, 4, 23, 256, 5319, 209868, 15912975, 2343052576, 675360194287, 383292136232380, 430038382710483623, 956430459603341708896, 4224538833207707658410103, 37106500399796746894085512140, 648740170822904504303462104598943

Offset: 0

Author

Gus Wiseman, Sep 04 2019

Keywords

• nonn

Crossrefs

The unlabeled version is A000088(n - 1).

Labeled graphs with no isolated vertices are A006129.

Covering graphs with at least one endpoint are A327227.

Cf. A006125, A006129, A054592, A245797, A327103, A327105.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      2^binomial(n, 2)-add(b(k)*binomial(n, k), k=0..n-1))
    end:
a:= n-> 2^(n*(n-1)/2)-b(n):
seq(a(n), n=0..17);  # Alois P. Heinz, Sep 04 2019

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#!=Range[n]&]],{n,0,5}]

PARI

b(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2)); \\ A006129
a(n) = 2^(n*(n-1)/2) - b(n); \\ Michel Marcus, Sep 05 2019

Formula

a(n) = A006125(n) - A006129(n).