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A327334
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Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity k.
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23
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1, 1, 0, 1, 1, 0, 4, 3, 1, 0, 26, 28, 9, 1, 0, 296, 490, 212, 25, 1, 0, 6064, 15336, 9600, 1692, 75, 1, 0, 230896, 851368, 789792, 210140, 14724, 231, 1, 0
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OFFSET
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0,7
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COMMENTS
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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).
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
1 1 0
4 3 1 0
26 28 9 1 0
296 490 212 25 1 0
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MATHEMATICA
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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]]]]]]]]];
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]}]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], vertConnSys[Range[n], #]==k&]], {n, 0, 5}, {k, 0, n}]
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CROSSREFS
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Row sums without the first two columns are A013922, if we assume A013922(1) = 0.
Spanning edge-connectivity is A327069.
Non-spanning edge-connectivity is A327148.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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