A372171 Number of labeled simple graphs covering n vertices with a unique triangle.

0, 0, 0, 1, 12, 220, 5460, 191975, 9596160, 683389812, 69270116040

Offset: 0

Author

Gus Wiseman, Apr 24 2024

Keywords

• nonn
• more

Comments

The unlabeled version is A372174.

Examples

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

Crossrefs

Column k = 1 of A372167, unlabeled A372173.

For no triangles we have A372168 (non-covering A213434), unlabeled A372169.

The non-covering case is A372172, unlabeled A372194.

The unlabeled version is A372174.

For all cycles (not just triangles) we have A372195, non-covering A372193.

A001858 counts acyclic graphs, unlabeled A005195.

A006125 counts simple graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494

A054548 counts labeled covering graphs by edges, unlabeled A370167.

A105784 counts acyclic covering graphs, unlabeled A144958.

A372170 counts graphs by triangles, unlabeled A263340.

A372175 counts covering graphs by cycles, non-covering A372176.

Cf. A000272, A053530, A121251, A137916, A367868, A369199, A372191.

Programs

Mathematica

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Union@@#==Range[n]&&Length[cys[#]]==1&]],{n,0,5}]

Formula

Inverse binomial transform of A372172.

Extensions

a(7)-a(10) from Andrew Howroyd, Aug 01 2024