A327200
Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.
Original entry on oeis.org
0, 0, 0, 4, 42, 718, 26262, 1878422, 256204460, 67525498676, 34969833809892, 35954978661632864, 73737437034063350534, 302166248212488958298674, 2475711390267267917290354410, 40563960064630744031043287569378, 1329219366981359393514586291328267704
Offset: 0
Row sums of
A327148 if the first two columns are removed.
BII-numbers of set-systems with non-spanning edge-connectivity >= 2 are
A327102.
Graphs with non-spanning edge-connectivity 1 are
A327231.
Cf.
A001187,
A006129,
A095983,
A182100,
A322395,
A326787,
A327076,
A327079,
A327097,
A327099,
A327236.
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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]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],eConn[#]>=2&]],{n,0,5}]
A327078
Binomial transform of A001187 (labeled connected graphs), if we assume A001187(1) = 0.
Original entry on oeis.org
1, 1, 2, 8, 61, 969, 31738, 2069964, 267270033, 68629753641, 35171000942698, 36024807353574280, 73784587576805254653, 302228602363365451957793, 2475873310144021668263093202, 40564787336902311168400640561084
Offset: 0
The a(0) = 1 through a(3) = 8 edge-sets:
{} {} {} {}
{{1,2}} {{1,2}}
{{1,3}}
{{2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
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b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-add(
k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
end:
a:= n-> add(b(n-j)*binomial(n, j), j=0..n-2)+1:
seq(a(n), n=0..18); # Alois P. Heinz, Aug 27 2019
-
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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]<=1&]],{n,0,5}]
Showing 1-2 of 2 results.
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