A327236 -id:A327236 - cp's OEIS frontend

A259862 Triangle read by rows: T(n,k) = number of unlabeled graphs with n nodes and connectivity exactly k (n>=1, 0<=k<=n-1).

1, 1, 1, 2, 1, 1, 5, 3, 2, 1, 13, 11, 7, 2, 1, 44, 56, 39, 13, 3, 1, 191, 385, 332, 111, 21, 3, 1, 1229, 3994, 4735, 2004, 345, 34, 4, 1, 13588, 67014, 113176, 66410, 13429, 992, 54, 4, 1, 288597, 1973029, 4629463, 3902344, 1109105, 99419, 3124, 81, 5, 1, 12297299, 105731474, 327695586, 388624106, 162318088, 21500415, 820956, 9813, 121, 5, 1

Offset: 1

N. J. A. Sloane, Jul 08 2015

These are vertex-connectivities. Spanning edge-connectivity is A263296. Non-spanning edge-connectivity is A327236. Cut-connectivity is A327127. - Gus Wiseman, Sep 03 2019

			Triangle begins:
       1;
       1,       1;
       2,       1,       1;
       5,       3,       2,       1;
      13,      11,       7,       2,       1;
      44,      56,      39,      13,       3,     1;
     191,     385,     332,     111,      21,     3,    1;
    1229,    3994,    4735,    2004,     345,    34,    4,  1;
   13588,   67014,  113176,   66410,   13429,   992,   54,  4, 1;
  288597, 1973029, 4629463, 3902344, 1109105, 99419, 3124, 81, 5, 1;
  12297299,105731474,327695586,388624106,162318088,21500415,820956,9813,121,5,1;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..78
Brendan McKay, confusion over k-connected graphs, posting to Sequence Fans Mailing List, Jul 08 2015.
Jens M. Schmidt, Combinatorial Data
Gus Wiseman, The graphs counted by row n = 5 (isolated vertices not shown).

Columns k=0..10 (up to initial nonzero terms) are A000719, A052442, A052443, A052444, A052445, A324234, A324235, A324088, A324089, A324090, A324091.
Row sums are A000088.
Number of graphs with connectivity at least k for k=1..10 are A001349, A002218, A006290, A086216, A086217, A324240, A324092, A324093, A324094, A324095.
The labeled version is A327334.
Cf. A002494, A263296, A322389, A326786, A327127, A327236.

A263296 Triangle read by rows: T(n,k) is the number of graphs with n vertices with edge connectivity k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 3, 2, 1, 13, 10, 8, 2, 1, 44, 52, 41, 15, 3, 1, 191, 351, 352, 121, 25, 3, 1, 1229, 3714, 4820, 2159, 378, 41, 4, 1, 13588, 63638, 113256, 68715, 14306, 1095, 65, 4, 1, 288597, 1912203, 4602039, 3952378, 1141575, 104829, 3441, 100, 5, 1

Offset: 1

Christian Stump, Oct 13 2015

This is spanning edge-connectivity. The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. The non-spanning edge-connectivity of a graph (A327236) is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty. Compare to vertex-connectivity (A259862). - Gus Wiseman, Sep 03 2019

			Triangle begins:
     1;
     1,    1;
     2,    1,    1;
     5,    3,    2,    1;
    13,   10,    8,    2,   1;
    44,   52,   41,   15,   3,  1;
   191,  351,  352,  121,  25,  3, 1;
  1229, 3714, 4820, 2159, 378, 41, 4, 1;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..78 (rows 1..12)
FindStat - Combinatorial Statistic Finder, The edge connectivity of a graph.
Jens M. Schmidt, Combinatorial Data
Gus Wiseman, Unlabeled graphs with 5 vertices organized by spanning edge-connectivity (isolated vertices not shown).

Row sums give A000088, n >= 1.
Columns k=0..10 are A000719, A052446, A052447, A052448, A241703, A241704, A241705, A324096, A324097, A324098, A324099.
Number of graphs with edge connectivity at least k for k=1..10 are A001349, A007146, A324226, A324227, A324228, A324229, A324230, A324231, A324232, A324233.
The labeled version is A327069.
Cf. A002494, A095983, A259862, A327076, A327108, A327109, A327111, A327144, A327145, A327147, A327236.

a(22)-a(55) added by Andrew Howroyd, Aug 11 2019

A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 4, 18, 27, 14, 1, 56, 250, 402, 240, 65, 10, 1, 1031, 5475, 11277, 9620, 4282, 921, 146, 15, 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 isolated vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   1
   1   1
   1   3   3   1
   4  18  27  14   1
  56 250 402 240  65  10   1

Row sums are A006125.
Column k = 0 is A327199.
Column k = 1 is A327231.
The corresponding triangle for vertex-connectivity is A327125.
The corresponding triangle for spanning edge-connectivity is A327069.
The covering version is A327149.
The unlabeled version is A327236, with covering version A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327079, A327097, A327099, A327102, A327126, A327144, A327196, A327200, A327201.

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

T(n,k) = Sum_{m = 0..n} binomial(n,m) A327149(m,k). In words, column k is the binomial transform of column k of A327149.

a(20)-a(28) from Robert Price, May 25 2021

A327201 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs covering n vertices with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 7, 5, 4, 1, 1

Offset: 0

Gus Wiseman, Sep 03 2019

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

			Triangle begins:
  1
  {}
  0 1
  0 0 1 1
  1 1 2 2 1
  2 3 7 5 4 1 1

Gus Wiseman, Unlabeled graphs covering 5 vertices, organized by non-spanning edge-connectivity.

Row sums are A002494.
Column k = 0 is A327075.
The labeled version is A327149.
Spanning edge-connectivity is A263296.
The non-covering version is A327236 (partial sums).
Cf. A000088, A322338, A322396, A326787, A327076, A327077, A327079, A327126, A327129, A327148, A327235.

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.

A327353 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 7, 3, 1, 53, 27, 45, 36, 6, 747, 511, 1497, 2085, 1540, 693, 316, 135, 45, 10, 1

Offset: 0

Gus Wiseman, Sep 10 2019

An antichain is a set of sets, none of which is a subset of any other.

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.

			Triangle begins:
    1
    1    1
    2    3
    8    7    3    1
   53   27   45   36    6
  747  511 1497 2085 1540  693  316  135   45   10    1
Row n = 3 counts the following antichains:
  {}             {{1}}      {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
  {{1},{2}}      {{2}}      {{1,2},{2,3}}
  {{1},{3}}      {{3}}      {{1,3},{2,3}}
  {{2},{3}}      {{1,2}}
  {{1},{2,3}}    {{1,3}}
  {{2},{1,3}}    {{2,3}}
  {{3},{1,2}}    {{1,2,3}}
  {{1},{2},{3}}

Row sums are A014466.
Column k = 0 is A327354.
The covering case is A327357.
The version for spanning edge-connectivity is A327352.
The specialization to simple graphs is A327148, with covering case A327149, unlabeled version A327236, and unlabeled covering case A327201.
Cf. A052446, A307249, A326704, A326787, A327071, A327351, A327355.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],eConn[#]==k&]],{n,0,4},{k,0,2^n}]//.{foe___,0}:>{foe}

A327354 Number of disconnected or empty antichains of nonempty subsets of {1..n} (non-spanning edge-connectivity 0).

Original entry on oeis.org

1, 1, 2, 8, 53, 747, 45156, 54804920, 19317457655317

Offset: 0

Gus Wiseman, Sep 10 2019

An antichain is a set of sets, none of which is a subset of any other.

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.

			The a(1) = 1 through a(3) = 8 antichains:
  {}  {}         {}
      {{1},{2}}  {{1},{2}}
                 {{1},{3}}
                 {{2},{3}}
                 {{1},{2,3}}
                 {{2},{1,3}}
                 {{3},{1,2}}
                 {{1},{2},{3}}

Column k = 0 of A327353.
The covering case is A120338.
The unlabeled version is A327426.
The spanning edge-connectivity version is A327352.
Cf. A014466, A326787, A327071, A327148, A327236, A327355, A327357.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Length[csm[#]]!=1&]],{n,0,4}]

Equals the binomial transform of the exponential transform of A048143 minus A048143.

A327199 Number of labeled simple graphs with n vertices whose edge-set is not connected.

Original entry on oeis.org

1, 1, 1, 1, 4, 56, 1031, 27189, 1165424, 89723096, 13371146135, 3989665389689, 2388718032951812, 2852540291841718752, 6768426738881535155247, 31870401029679493862010949, 297787425565749788134314214272

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

Also graphs with non-spanning edge-connectivity 0.

			The a(4) = 4 edge-sets: {}, {12,34}, {13,24}, {14,23}.

Column k = 0 of A327148.
The covering case is A327070.
The unlabeled version is A327235.
Cf. A001187, A006129, A322395, A326787, A327075, A327076, A327079, A327129, A327200, A327201, A327231, 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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]!=1&]],{n,0,5}]

Binomial transform of A327070.

A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 0, 1, 1, 1, 4, 1, 3, 1, 30, 13, 33, 32, 6, 546, 421, 1302, 1915, 1510, 693, 316, 135, 45, 10, 1

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

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.

			Triangle begins:
    1
    0    1
    1    1
    4    1    3    1
   30   13   33   32    6
  546  421 1302 1915 1510  693  316  135   45   10    1
Row n = 3 counts the following antichains:
  {{1},{2,3}}    {{1,2,3}}  {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
  {{2},{1,3}}               {{1,2},{2,3}}
  {{3},{1,2}}               {{1,3},{2,3}}
  {{1},{2},{3}}

Row sums are A307249.
Column k = 0 is A120338.
The non-covering version is A327353.
The version for spanning edge-connectivity is A327352.
The specialization to simple graphs is A327149, with unlabeled version A327201.
Cf. A014466, A293606, A326704, A326787, A327071, A327148, A327236, A327351.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,5},{k,0,2^n}]//.{foe___,0}:>{foe}

A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

Original entry on oeis.org

1, 1, 3, 6, 15, 52, 410, 32697

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 antichains:
  {}  {}         {}             {}
      {{1}}      {{1}}          {{1}}
      {{1},{2}}  {{1,2}}        {{1,2}}
                 {{1},{2}}      {{1},{2}}
                 {{1},{2,3}}    {{1,2,3}}
                 {{1},{2},{3}}  {{1},{2,3}}
                                {{1,2},{1,3}}
                                {{1},{2},{3}}
                                {{1},{2,3,4}}
                                {{1,2},{3,4}}
                                {{1},{2},{3,4}}
                                {{1},{2},{3},{4}}
                                {{2},{1,3},{1,4}}
                                {{1,2},{1,3},{2,3}}
                                {{4},{1,2},{1,3},{2,3}}

Column k = 0 of A327438.
The labeled version is A327355.
The covering case is A327426.
Cf. A014466 A052446, A120338, A261005, A293606, A327201, A327236, A327352, A327353, A327354, A327438.

a(n > 0) = A306505(n) - A261006(n).

cp's OEIS Frontend

A259862 Triangle read by rows: T(n,k) = number of unlabeled graphs with n nodes and connectivity exactly k (n>=1, 0<=k<=n-1).

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A263296 Triangle read by rows: T(n,k) is the number of graphs with n vertices with edge connectivity k.

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A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

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A327201 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs covering n vertices with non-spanning edge-connectivity k.

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

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A327353 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.

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A327354 Number of disconnected or empty antichains of nonempty subsets of {1..n} (non-spanning edge-connectivity 0).

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A327199 Number of labeled simple graphs with n vertices whose edge-set is not connected.

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A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

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