

A330297


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


5




OFFSET

0,4


COMMENTS

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


LINKS

Table of n, a(n) for n=0..6.
Gus Wiseman, All 9 distinct unlabeled representatives of the a(5) = 540 graphs.


FORMULA

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


EXAMPLE

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}


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}]


CROSSREFS

The noncovering 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.
Sequence in context: A336577 A194157 A166736 * A109055 A318766 A292813
Adjacent sequences: A330294 A330295 A330296 * A330298 A330299 A330300


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Dec 12 2019


STATUS

approved



