A327363 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity >= k.

1, 1, 0, 2, 1, 0, 8, 4, 1, 0, 64, 38, 10, 1, 0, 1024, 728, 238, 26, 1, 0

Offset: 0

Author

Gus Wiseman, Sep 26 2019

Keywords

• nonn
• tabl
• more

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.

Examples

			Triangle begins:
     1
     1    0
     2    1    0
     8    4    1    0
    64   38   10    1    0
  1024  728  238   26    1    0
		

Crossrefs

Column k = 0 is A006125.

Column k = 1 is A001187.

Column k = 2 is A013922.

The unlabeled version is A327805.

Row-wise partial sums of A327334 (vertex-connectivity exactly k).

Cf. A259862, A327114, A327125, A327126, A327127, A327806.

Programs

Mathematica

vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]