A330297 Number of labeled simple graphs covering n vertices with exactly two automorphisms, or with exactly n!/2 graphs obtainable by permuting the vertices.

0, 0, 1, 3, 24, 540, 13320

Offset: 0

Author

Gus Wiseman, Dec 12 2019

Keywords

• nonn
• more

Comments

These are graphs with exactly one involution and no other symmetries.

Examples

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

Links

• Gus Wiseman, All 9 distinct unlabeled representatives of the a(5) = 540 graphs.

Crossrefs

The non-covering version is A330345.

The unlabeled version is A330346 (not A241454).

Covering simple graphs are A006129.

Covering graphs with exactly one automorphism are A330343.

Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), and A330346 (unlabeled covering).

Cf. A003400, A006125, A016031, A124059, A143543, A241454, A330098, A330229, A330230, A330231, A330233.

Programs

Mathematica

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[graprms[#]]==n!/2&]],{n,0,5}]

Formula

a(n) = n!/2 * A330346(n).