A327237 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices that, if the isolated vertices are removed, have cut-connectivity k.

1, 1, 0, 1, 0, 1, 1, 3, 3, 1, 4, 40, 15, 4, 1, 56, 660, 267, 35, 5, 1, 1031, 18756, 11022, 1862, 90, 6, 1

Offset

0,8

Comments

We define the cut-connectivity of a graph to be the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph, with the exception that a graph with one vertex has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.

Links

Table of n, a(n) for n=0..27.

Formula

Column-wise binomial transform of A327126.

Example

Triangle begins:
   1
   1   0
   1   0   1
   1   3   3   1
   4  40  15   4   1
  56 660 267  35   5   1

Mathematica

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_, eds_]:=If[Length[vts]==1, 1, Min@@Length/@Select[Subsets[vts], Function[del, csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], cutConnSys[Union@@#, #]==k&]], {n, 0, 4}, {k, 0, n}]

Crossrefs

Row sums are A006125.

Column k = 0 is A327199.

The covering case is A327126.

Row sums without the first column are A287689.

Cf. A006125, A001187, A013922, A259862, A322389, A326786, A327070, A327114, A327125, A327127, A327198.

Sequence in context: A111408 A327148 A366595 * A092674 A316366 A111945

Adjacent sequences: A327234 A327235 A327236 * A327238 A327239 A327240

Keyword

nonn,tabl,more

Author

Gus Wiseman, Sep 03 2019

Extensions

a(21)-a(27) from Jinyuan Wang, Jun 27 2020

Status

approved