A372172 - OEIS

%I #9 Aug 01 2024 01:21:34

%S 0,0,0,1,16,290,6980,235270,11298056,777154308,76560083040

%N Number of labeled simple graphs on n vertices with exactly one triangle.

%C The unlabeled version is A372194.

%F Binomial transform of A372171.

%e The a(4) = 16 graphs:

%e   12,13,23

%e   12,14,24

%e   13,14,34

%e   23,24,34

%e   12,13,14,23

%e   12,13,14,24

%e   12,13,14,34

%e   12,13,23,24

%e   12,13,23,34

%e   12,14,23,24

%e   12,14,24,34

%e   12,23,24,34

%e   13,14,23,34

%e   13,14,24,34

%e   13,23,24,34

%e   14,23,24,34

%t cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cys[#]]==1&]],{n,0,5}]

%Y For no triangles we have A213434, covering A372168 (unlabeled A372169).

%Y Column k = 1 of A372170, unlabeled A263340.

%Y The covering case is A372171, unlabeled A372174.

%Y For all cycles (not just triangles) we have A372193, covering A372195.

%Y The unlabeled version is A372194.

%Y A001858 counts acyclic graphs, unlabeled A005195.

%Y A006125 counts simple graphs, unlabeled A000088.

%Y A006129 counts covering graphs, unlabeled A002494

%Y A054548 counts labeled covering graphs by edges, unlabeled A370167.

%Y A372167 counts covering graphs by triangles, unlabeled A372173.

%Y Cf. A000272, A105784, A121251, A137916, A236570, A372176.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Apr 24 2024

%E a(8)-a(10) from _Andrew Howroyd_, Aug 01 2024