A327125 - OEIS

%I #16 May 20 2021 10:54:48

%S 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 1692,75,0,1,230896

%N Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and 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   1

%e     1   0   1

%e     4   3   0   1

%e    26  28   9   0   1

%e   296 490 212  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}]],cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

%Y After the first column, same as A327126.

%Y The unlabeled version is A327127.

%Y Row sums are A006125.

%Y Column k = 0 is A054592, if we assume A054592(0) = 1.

%Y Column k = 1 is A327114, if we assume A327114(1) = 1.

%Y Row sums without the first column are A001187.

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

%Y Different from A327069.

%Y Cf. A259862, A322389, A326786, A327082, A327098, A327100, A327101.

%K nonn,more,tabl

%O 0,7

%A _Gus Wiseman_, Aug 25 2019

%E a(21)-a(28) from _Robert Price_, May 20 2021

%E a(1) and a(2) corrected by _Robert Price_, May 20 2021