A322555 Number of labeled simple graphs on n vertices where all non-isolated vertices have the same degree.

1, 1, 2, 5, 18, 69, 390, 2703, 59474, 1548349, 168926258, 12165065351, 7074423247562, 2294426405580191, 4218009215702391954, 3810376434461484994317, 35102248193591661086921250, 156873334244228518638713087133, 4144940994226400702145709978234154

Offset: 0

Author

Gus Wiseman, Dec 15 2018

Keywords

• nonn

Comments

Such graphs may be said to have regular support.

Examples

			The a(4) = 18 edge sets:
  {}
  {{1,2}}
  {{1,3}}
  {{1,4}}
  {{2,3}}
  {{2,4}}
  {{3,4}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
		

Crossrefs

Cf. A000569, A005176, A038041, A059441, A295193, A306017, A306019, A319169, A320458.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],SameQ@@Length/@Split[Sort[Join@@#]]&]],{n,6}]

Formula

a(n) = 1 + Sum_{k=1..n} binomial(n, k)*(A295193(k) - 1). - Andrew Howroyd, Dec 17 2018

Extensions

a(8)-a(15) from Andrew Howroyd, Dec 17 2018

a(16)-a(18) from Andrew Howroyd, May 21 2020