A327126 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with cut-connectivity k.

1, 0, 0, 0, 0, 1, 0, 3, 0, 1, 3, 28, 9, 0, 1, 40, 490, 212, 25, 0, 1, 745, 15336, 9600, 1692, 75, 0, 1

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   0
   0   0   1
   0   3   0   1
   3  28   9   0   1
  40 490 212  25   0   1
		

Crossrefs

After the first column, same as A327125.

Column k = 0 is A327070.

Column k = 1 is A327114.

Row sums are A006129.

Different from A327069.

Row sums without the first column are A001187, if we assume A001187(0) = A001187(1) = 0.

Row sums without the first two columns are A013922.

Cf. A006125, A259862, A322389, A326786, A327114, A327127, A327198, A327237.

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}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

Extensions

a(21)-a(27) from Robert Price, May 20 2021