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A327125 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and cut-connectivity k.

%I A327125 #16 May 20 2021 10:54:48
%S A327125 1,0,1,1,0,1,4,3,0,1,26,28,9,0,1,296,490,212,25,0,1,6064,15336,9600,
%T A327125 1692,75,0,1,230896
%N A327125 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and cut-connectivity k.
%C A327125 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 A327125 Triangle begins:
%e A327125     1
%e A327125     0   1
%e A327125     1   0   1
%e A327125     4   3   0   1
%e A327125    26  28   9   0   1
%e A327125   296 490 212  25   0   1
%t A327125 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 A327125 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 A327125 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]
%Y A327125 After the first column, same as A327126.
%Y A327125 The unlabeled version is A327127.
%Y A327125 Row sums are A006125.
%Y A327125 Column k = 0 is A054592, if we assume A054592(0) = 1.
%Y A327125 Column k = 1 is A327114, if we assume A327114(1) = 1.
%Y A327125 Row sums without the first column are A001187.
%Y A327125 Row sums without the first two columns are A013922.
%Y A327125 Different from A327069.
%Y A327125 Cf. A259862, A322389, A326786, A327082, A327098, A327100, A327101.
%K A327125 nonn,more,tabl
%O A327125 0,7
%A A327125 _Gus Wiseman_, Aug 25 2019
%E A327125 a(21)-a(28) from _Robert Price_, May 20 2021
%E A327125 a(1) and a(2) corrected by _Robert Price_, May 20 2021