A327125 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and cut-connectivity k.

1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 26, 28, 9, 0, 1, 296, 490, 212, 25, 0, 1, 6064, 15336, 9600, 1692, 75, 0, 1, 230896

Offset

0,7

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 and no edges has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.

Links

Table of n, a(n) for n=0..28.

Example

Triangle begins:
    1
    0   1
    1   0   1
    4   3   0   1
   26  28   9   0   1
  296 490 212  25   0   1

Mathematica

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, 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[Range[n], #]==k&]], {n, 0, 4}, {k, 0, n}]

Crossrefs

After the first column, same as A327126.

The unlabeled version is A327127.

Row sums are A006125.

Column k = 0 is A054592, if we assume A054592(0) = 1.

Column k = 1 is A327114, if we assume A327114(1) = 1.

Row sums without the first column are A001187.

Row sums without the first two columns are A013922.

Different from A327069.

Cf. A259862, A322389, A326786, A327082, A327098, A327100, A327101.

Sequence in context: A331956 A325019 A152151 * A152148 A270708 A198261

Adjacent sequences: A327122 A327123 A327124 * A327126 A327127 A327128

Keyword

nonn,more,tabl

Author

Gus Wiseman, Aug 25 2019

Extensions

a(21)-a(28) from Robert Price, May 20 2021

a(1) and a(2) corrected by Robert Price, May 20 2021

Status

approved