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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.
18
1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 26, 28, 9, 0, 1, 296, 490, 212, 25, 0, 1, 6064, 15336, 9600, 1692, 75, 0, 1, 230896
OFFSET
0,7
COMMENTS
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.
EXAMPLE
Triangle begins:
1
0 1
1 0 1
4 3 0 1
26 28 9 0 1
296 490 212 25 0 1
MATHEMATICA
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]]]]]]]]];
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]}]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], cutConnSys[Range[n], #]==k&]], {n, 0, 4}, {k, 0, n}]
CROSSREFS
After the first column, same as A327126.
The unlabeled version is A327127.
Row sums are A006125.
Column k = 0 is A054592, if we assume A054592(0) = 1.
Column k = 1 is A327114, if we assume A327114(1) = 1.
Row sums without the first column are A001187.
Row sums without the first two columns are A013922.
Different from A327069.
Sequence in context: A331956 A325019 A152151 * A152148 A270708 A198261
KEYWORD
nonn,more,tabl
AUTHOR
Gus Wiseman, Aug 25 2019
EXTENSIONS
a(21)-a(28) from Robert Price, May 20 2021
a(1) and a(2) corrected by Robert Price, May 20 2021
STATUS
approved