A327099 -id:A327099 - cp's OEIS frontend

A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.

1, 0, 1, 0, 0, 3, 1, 3, 12, 15, 10, 1, 40, 180, 297, 180, 60, 10, 1

Offset: 0

Gus Wiseman, Aug 27 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   {}
   0   1
   0   0   3   1
   3  12  15  10   1
  40 180 297 180  60  10   1

Row sums are A006129.
Column k = 0 is A327070.
Column k = 1 is A327079.
The corresponding triangle for vertex-connectivity is A327126.
The corresponding triangle for spanning edge-connectivity is A327069.
The non-covering version is A327148.
The unlabeled version is A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327097, A327099, A327102, A327125, A327129, A327144.

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]]]]]]]]];
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}]],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

A327148(n,k) = Sum_{m = 0..n} binomial(n,m) T(m,k). In words, column k is the inverse binomial transform of column k of A327148.

A327231 Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 0, 1, 3, 18, 250, 5475, 191541, 11065572, 1104254964, 201167132805, 69828691941415, 47150542741904118, 62354150876493659118, 161919876753750972738791, 827272271567137357352991705, 8331016130913639432634637862600, 165634930763383717802534343776893928

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

A bridge is an edge whose removal disconnected the graph, while an endpoint is a vertex belonging to only one edge. The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.

			The a(2) = 1 through a(4) = 18 edge-sets:
  {12}  {12}  {12}
        {13}  {13}
        {23}  {14}
              {23}
              {24}
              {34}
              {12,13,24}
              {12,13,34}
              {12,14,23}
              {12,14,34}
              {12,23,34}
              {12,24,34}
              {13,14,23}
              {13,14,24}
              {13,23,24}
              {13,24,34}
              {14,23,24}
              {14,23,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k = 1 of A327148.
The covering version is A327079.
Connected bridged graphs (spanning edge-connectivity 1) are A327071.
BII-numbers of set-systems with non-spanning edge-connectivity 1 are A327099.
Covering set-systems with non-spanning edge-connectivity 1 are A327129.
Cf. A001187, A052446, A322395, A327072, A327073, A327102.

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]]]]]]]]];
edgeConnSys[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}]],edgeConnSys[#]==1&]],{n,0,4}]

Binomial transform of A327079.

Terms a(6) and beyond from Andrew Howroyd, Sep 11 2019

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

Gus Wiseman, Sep 01 2019

nonn

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.

Gus Wiseman, The a(4) = 42 graphs with non-spanning edge-connectivity >= 2.

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.

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}]

Binomial transform of A322395, if we assume A322395(0) = A322395(1) = A322395(2) = 0.

A327196 Number of connected set-systems with n vertices and at least one bridge that is not an endpoint (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 1, 4, 44, 2960

Offset: 0

Gus Wiseman, Aug 31 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

			Non-isomorphic representatives of the a(3) = 44 set-systems:
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{1},{2},{1,2}}
  {{1},{1,2},{2,3}}
  {{1},{2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,2,3}}

The covering version is A327129.
The BII-numbers of these set-systems are A327099.
The restriction to simple graphs is A327231.
Set-systems with spanning edge-connectivity 1 are A327145.
Cf. A263296, A322395, A327079, A327111, A327148, A327149, A327200.

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],{1,n}]],eConn[#]==1&]],{n,0,3}]

Binomial transform of A327129.

cp's OEIS Frontend

A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.

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A327231 Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1).

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A327200 Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.

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A327196 Number of connected set-systems with n vertices and at least one bridge that is not an endpoint (non-spanning edge-connectivity 1).

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