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

Author

Gus Wiseman, Aug 25 2019

Keywords

• nonn
• more
• tabl

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.

Examples

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

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.

Programs

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}]

Extensions

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

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