A327334 - OEIS

%I #12 May 17 2021 04:36:43

%S 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 1692,75,1,0,230896,851368,789792,210140,14724,231,1,0

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

%C 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-vertex-connected_graph">k-vertex-connected graph</a>

%e Triangle begins:

%e     1

%e     1   0

%e     1   1   0

%e     4   3   1   0

%e    26  28   9   1   0

%e   296 490 212  25   1   0

%t 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 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 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]==k&]],{n,0,5},{k,0,n}]

%Y The unlabeled version is A259862.

%Y Row sums are A006125.

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

%Y Column k = 1 is A327336.

%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, if we assume A013922(1) = 0.

%Y Cut-connectivity is A327125.

%Y Spanning edge-connectivity is A327069.

%Y Non-spanning edge-connectivity is A327148.

%Y Cf. A322389, A327051, A327070, A327126, A327127, A327350.

%K nonn,tabl,more

%O 0,7

%A _Gus Wiseman_, Sep 01 2019

%E a(21)-a(35) from _Robert Price_, May 14 2021