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A327237
Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices that, if the isolated vertices are removed, have cut-connectivity k.
4
1, 1, 0, 1, 0, 1, 1, 3, 3, 1, 4, 40, 15, 4, 1, 56, 660, 267, 35, 5, 1, 1031, 18756, 11022, 1862, 90, 6, 1
OFFSET
0,8
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 has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.
FORMULA
Column-wise binomial transform of A327126.
EXAMPLE
Triangle begins:
1
1 0
1 0 1
1 3 3 1
4 40 15 4 1
56 660 267 35 5 1
MATHEMATICA
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]]]]]]]]];
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[Union@@#, #]==k&]], {n, 0, 4}, {k, 0, n}]
CROSSREFS
Row sums are A006125.
Column k = 0 is A327199.
The covering case is A327126.
Row sums without the first column are A287689.
Sequence in context: A111408 A327148 A366595 * A092674 A316366 A111945
KEYWORD
nonn,tabl,more
AUTHOR
Gus Wiseman, Sep 03 2019
EXTENSIONS
a(21)-a(27) from Jinyuan Wang, Jun 27 2020
STATUS
approved