A236570 Number of n-node simple unicyclic graphs.

1, 3, 9, 25, 68, 185, 504, 1379, 3788, 10480, 29094, 81193, 227379, 639099, 1801394, 5091388, 14422301, 40939337, 116420959, 331622137, 946020596, 2702412657, 7729367873, 22132856218, 63444473053, 182046034559, 522841943138, 1502920139133

Offset: 3

Author

Eric W. Weisstein, Jan 29 2014

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..500
• Eric Weisstein's World of Mathematics, Unicyclic Graph.
• Gus Wiseman, The a(6) = 25 graphs with a unique cycle.

Crossrefs

The covering version is A372191, labeled A372195.

The labeled version is A372193.

Cf. A001429 (number of connected n-node unicyclic graphs), A005195.

Programs

Mathematica

Needs["Combinatorica`"];nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
b = Drop[Flatten[
   sol = SolveAlways[
     0 == Series[
       t[x] - x Product[1/(1 - x^i)^ a[i], {i, 1, nn}], {x, 0, nn}],
     x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
    CoefficientList[
     Series[CycleIndex[DihedralGroup[n], s] /.
       Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
     nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; f =
Drop[CoefficientList[Series[r[x] - (r[x]^2 - r[x^2])/2, {x, 0, nn}],
   x], 1]; Drop[CoefficientList[
Series[d[x] Product[1/(1 - x^i)^f[[i]], {i, 1, nn}], {x, 0, nn}], x],3] (* Geoffrey Critzer, Nov 16 2014 *)

Formula

G.f.: A(x)*B(x) where A(x) is the o.g.f. for A001429 and B(x) is the o.g.f. for A005195. - Geoffrey Critzer, Nov 16 2014

Partial sums of A372191. - Gus Wiseman, Apr 27 2024

Extensions

a(11)-a(30) from Geoffrey Critzer, Nov 16 2014