A380805 -id:A380805 - cp's OEIS frontend

A380632 Number of simple connected graphs on n unlabeled nodes with each node a member of exactly one cycle.

1, 0, 0, 1, 1, 1, 2, 2, 3, 5, 9, 14, 28, 49, 95, 182, 369, 733, 1509, 3103, 6504, 13627, 28949, 61701, 132457, 285454, 618863, 1346022, 2940287, 6444364, 14172744, 31257883, 69142445, 153333476, 340880766, 759549740, 1696122213, 3795178540, 8508326129, 19109193805, 42991993545, 96881110654

Offset: 0

Andrew Howroyd, Feb 24 2025

nonn

All such graphs are cactus graphs (with bridges allowed).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Cactus graph.
Index entries for sequences related to cacti.

Row sums of A380631.

Cf. A000083, A380805.

PARI
```
Vec(G(40)) \\ G() defined in A380631.
```

A380633 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes of degree at most 3 with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 3, 0, 1, 3, 6, 0, 1, 4, 11, 2, 0, 1, 4, 17, 5, 0, 1, 5, 26, 17, 0, 1, 5, 36, 37, 2, 0, 1, 6, 50, 78, 12, 0, 1, 6, 65, 140, 44, 0, 1, 7, 85, 248, 131, 4, 0, 1, 7, 106, 396, 325, 23

Offset: 0

Andrew Howroyd, Feb 24 2025

All such graphs are cactus graphs (with bridges allowed).

			Triangle begins:
  1;
  0;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  3;
  0, 1, 3,  6;
  0, 1, 4, 11,  2;
  0, 1, 4, 17,  5;
  0, 1, 5, 26, 17;
  0, 1, 5, 36, 37, 2;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)
Wikipedia, Cactus graph.
Index entries for sequences related to cacti.

Columns 0..3 are A000007, A000012(n+3), A004526(n+4), A003453(n+4).
Row sums are A380805.
Cf. A000672, A380631 (with vertices of any degree).

raise(p,d) = {my(n=serprec(p,x)-1); substvec(p + O(x^(n\d+1)), [x, y], [x^d,y^d])}
R(n,y)={my(g=O(x^3)); for(n=1, (n-1)\2, my(p=x*(1 + g), p2=raise(p,2)); g=x*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
G(n,y=1)={my(g=R(n,y), p = x*(1+g) + O(x*x^n));
  my( r=((1 + p)^2/(1 - raise(p,2)) - 1)/2 );
  my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p,d))) );
  1 + (raise(g,2) - g^2 + y*(r + c - 2*p - p^2 - raise(p,2)))/2 }
T(n)={[Vecrev(p) | p<-Vec(G(n,y))]}
{my(A=T(15)); for(i=1, #A, print(A[i]))}

T(3*n,n) = A000672(n).

A381468 Number of simple connected graphs on n unlabeled nodes with no node a member of more than one cycle.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 20, 48, 133, 374, 1124, 3439, 10923, 35245, 116128, 387729, 1312038, 4485906, 15486546, 53900520, 188998450, 667062919, 2368440477, 8454560144, 30328595227, 109285433191, 395425965732, 1436219868659, 5234881134074, 19143123415166, 70216752517419

Offset: 0

Andrew Howroyd, Feb 24 2025

nonn

All such graphs are cactus graphs (with bridges allowed).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
R. J. Mathar, Counting connected graphs without overlapping cycles, arXiv:1808.06264 [math.CO] (2018).
Wikipedia, Cactus graph.
Index entries for sequences related to cacti.

Row sums of A381467.

Cf. A000083, A317722 (with 2-cycles), A380632, A380805.

PARI
```
Vec(G(31,1)) \\ G() defined in A381467.
```

cp's OEIS Frontend

A380632 Number of simple connected graphs on n unlabeled nodes with each node a member of exactly one cycle.

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A380633 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes of degree at most 3 with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).

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A381468 Number of simple connected graphs on n unlabeled nodes with no node a member of more than one cycle.

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PARI