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A327334 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity k.

%I A327334 #12 May 17 2021 04:36:43
%S A327334 1,1,0,1,1,0,4,3,1,0,26,28,9,1,0,296,490,212,25,1,0,6064,15336,9600,
%T A327334 1692,75,1,0,230896,851368,789792,210140,14724,231,1,0
%N A327334 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity k.
%C A327334 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. Except for complete graphs, this is the same as cut-connectivity (A327125).
%H A327334 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-vertex-connected_graph">k-vertex-connected graph</a>
%e A327334 Triangle begins:
%e A327334     1
%e A327334     1   0
%e A327334     1   1   0
%e A327334     4   3   1   0
%e A327334    26  28   9   1   0
%e A327334   296 490 212  25   1   0
%t A327334 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 A327334 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]}]];
%t A327334 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]==k&]],{n,0,5},{k,0,n}]
%Y A327334 The unlabeled version is A259862.
%Y A327334 Row sums are A006125.
%Y A327334 Column k = 0 is A054592, if we assume A054592(0) = A054592(1) = 1.
%Y A327334 Column k = 1 is A327336.
%Y A327334 Row sums without the first column are A001187, if we assume A001187(0) = A001187(1) = 0.
%Y A327334 Row sums without the first two columns are A013922, if we assume A013922(1) = 0.
%Y A327334 Cut-connectivity is A327125.
%Y A327334 Spanning edge-connectivity is A327069.
%Y A327334 Non-spanning edge-connectivity is A327148.
%Y A327334 Cf. A322389, A327051, A327070, A327126, A327127, A327350.
%K A327334 nonn,tabl,more
%O A327334 0,7
%A A327334 _Gus Wiseman_, Sep 01 2019
%E A327334 a(21)-a(35) from _Robert Price_, May 14 2021