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A327237 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices that, if the isolated vertices are removed, have cut-connectivity k.

%I A327237 #9 Jun 27 2020 03:44:33
%S A327237 1,1,0,1,0,1,1,3,3,1,4,40,15,4,1,56,660,267,35,5,1,1031,18756,11022,
%T A327237 1862,90,6,1
%N A327237 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices that, if the isolated vertices are removed, have cut-connectivity k.
%C A327237 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 has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.
%F A327237 Column-wise binomial transform of A327126.
%e A327237 Triangle begins:
%e A327237    1
%e A327237    1   0
%e A327237    1   0   1
%e A327237    1   3   3   1
%e A327237    4  40  15   4   1
%e A327237   56 660 267  35   5   1
%t A327237 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327237 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 A327237 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],cutConnSys[Union@@#,#]==k&]],{n,0,4},{k,0,n}]
%Y A327237 Row sums are A006125.
%Y A327237 Column k = 0 is A327199.
%Y A327237 The covering case is A327126.
%Y A327237 Row sums without the first column are A287689.
%Y A327237 Cf. A006125, A001187, A013922, A259862, A322389, A326786, A327070, A327114, A327125, A327127, A327198.
%K A327237 nonn,tabl,more
%O A327237 0,8
%A A327237 _Gus Wiseman_, Sep 03 2019
%E A327237 a(21)-a(27) from _Jinyuan Wang_, Jun 27 2020