A326240 - OEIS

%I #4 Jun 17 2019 21:47:26

%S 0,2,0,8,160,6976,644992

%N Number of Hamiltonian labeled n-vertex graphs with loops.

%C A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once.

%F a(n) = A326208(n) * 2^n.

%e The a(3) = 8 edge-sets:

%e   {12,13,23}  {11,12,13,23}  {11,12,13,22,23}  {11,12,13,22,23,33}

%e               {12,13,22,23}  {11,12,13,23,33}

%e               {12,13,23,33}  {12,13,22,23,33}

%t Table[Length[Select[Subsets[Select[Tuples[Range[n],2],OrderedQ]],FindHamiltonianCycle[Graph[Range[n],#]]!={}&]],{n,0,5}]

%Y The unlabeled case is A326215.

%Y The directed case is A326204 (with loops) or A326219 (without loops).

%Y The case without loops A326208.

%Y Graphs with loops not containing a Hamiltonian cycle are A326239.

%Y Cf. A000088, A003216, A006125, A057864, A283420.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Jun 17 2019