A327126 - OEIS

%I #17 May 20 2021 22:59:12

%S 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,

%T 75,0,1

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

%C 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.

%e Triangle begins:

%e    1

%e    0   0

%e    0   0   1

%e    0   3   0   1

%e    3  28   9   0   1

%e   40 490 212  25   0   1

%t 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]]]]]]]]];

%t 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]}]]];

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

%Y After the first column, same as A327125.

%Y Column k = 0 is A327070.

%Y Column k = 1 is A327114.

%Y Row sums are A006129.

%Y Different from A327069.

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

%Y Row sums without the first two columns are A013922.

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

%K nonn,more,tabl

%O 0,8

%A _Gus Wiseman_, Aug 25 2019

%E a(21)-a(27) from _Robert Price_, May 20 2021