A327805 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.

1, 1, 0, 2, 1, 0, 4, 2, 1, 0, 11, 6, 3, 1, 0, 34, 21, 10, 3, 1, 0, 156, 112, 56, 17, 4, 1, 0, 1044, 853, 468, 136, 25, 4, 1, 0, 12346, 11117, 7123, 2388, 384, 39, 5, 1, 0, 274668, 261080, 194066, 80890, 14480, 1051, 59, 5, 1, 0, 12005168, 11716571, 9743542, 5114079, 1211735, 102630, 3211, 87, 6, 1, 0

Offset: 0

Author

Gus Wiseman, Sep 26 2019

Keywords

• nonn
• tabl

Comments

The vertex-connectivity of a graph is the minimum number of vertices that must be removed (along with any incident edges) to obtain a non-connected graph or singleton. Note that this means a single node has vertex-connectivity 0.

Examples

			Triangle begins:
   1
   1  0
   2  1  0
   4  2  1  0
  11  6  3  1  0
  34 21 10  3  1  0
		

Links

• Gus Wiseman, The graphs counted in row n = 4 (isolated vertices not shown).

Crossrefs

Row-wise partial sums of A259862.

The labeled version is A327363.

The covering case is A327365, from which this sequence differs only in the k = 0 column.

Column k = 0 is A000088 (graphs).

Column k = 1 is A001349 (connected graphs), if we assume A001349(0) = A001349(1) = 0.

Column k = 2 is A002218 (2-connected graphs), if we assume A002218(2) = 0.

The triangle for vertex-connectivity exactly k is A259862.

Cf. A326786, A327051, A327114, A327125, A327126, A327127, A327334.

Formula

T(n,k) = Sum_{j=k..n} A259862(n,j).

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 26 2020