A259862 -id:A259862 - cp's OEIS frontend

A052444 Number of simple unlabeled n-node graphs of connectivity 3.

Original entry on oeis.org

0, 0, 0, 1, 2, 13, 111, 2004, 66410, 3902344, 388624106, 65142804740

Offset: 1

Eric W. Weisstein

Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph.

Column k=3 of A259862.

Cf. A000719, A006290, A052442, A052443, A052445, A086216.

a(n) = A006290(n) - A086216(n). - Andrew Howroyd, Sep 04 2019

Name edited and a(8)-a(11) by Jens M. Schmidt, Feb 18 2019
a(3)-a(4) corrected by Andrew Howroyd, Aug 28 2019
a(12) from Sean A. Irvine, Nov 28 2021

A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1.

Original entry on oeis.org

0, 1, 2, 27, 2084

Offset: 0

Gus Wiseman, Sep 02 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. We define the cut-connectivity (A326786, A327237, A327126) of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).

The covering version is A327197.
The BII-numbers of these set-systems are A327098.
Cf. A003465, A052442, A052443, A259862, A323818, A326786, A327101, A327112, A327113, A327114, A327126, A327229.

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],{1,n}]],cutConnSys[Union@@#,#]==1&]],{n,0,3}]

Binomial transform of A327197.

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.

Original entry on oeis.org

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

Gus Wiseman, Sep 03 2019

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.

			Triangle begins:
   1
   1   0
   1   0   1
   1   3   3   1
   4  40  15   4   1
  56 660 267  35   5   1

Row sums are A006125.
Column k = 0 is A327199.
The covering case is A327126.
Row sums without the first column are A287689.
Cf. A006125, A001187, A013922, A259862, A322389, A326786, A327070, A327114, A327125, A327127, A327198.

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}]

Column-wise binomial transform of A327126.

a(21)-a(27) from Jinyuan Wang, Jun 27 2020

A327805 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 2, 1, 0, 11, 6, 3, 1, 0, 34, 21, 10, 3, 1, 0, 156, 112, 56, 17, 4, 1, 0, 1044, 853, 468, 136, 25, 4, 1, 0, 12346, 11117, 7123, 2388, 384, 39, 5, 1, 0, 274668, 261080, 194066, 80890, 14480, 1051, 59, 5, 1, 0, 12005168, 11716571, 9743542, 5114079, 1211735, 102630, 3211, 87, 6, 1, 0

Offset: 0

Gus Wiseman, Sep 26 2019

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. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
   1
   1  0
   2  1  0
   4  2  1  0
  11  6  3  1  0
  34 21 10  3  1  0

Gus Wiseman, The graphs counted in row n = 4 (isolated vertices not shown).

Row-wise partial sums of A259862.
The labeled version is A327363.
The covering case is A327365, from which this sequence differs only in the k = 0 column.
Column k = 0 is A000088 (graphs).
Column k = 1 is A001349 (connected graphs), if we assume A001349(0) = A001349(1) = 0.
Column k = 2 is A002218 (2-connected graphs), if we assume A002218(2) = 0.
The triangle for vertex-connectivity exactly k is A259862.
Cf. A326786, A327051, A327114, A327125, A327126, A327127, A327334.

T(n,k) = Sum_{j=k..n} A259862(n,j).

Terms a(21) and beyond from Andrew Howroyd, Dec 26 2020

A086217 Number of 5-connected graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 39, 1051, 102630, 22331311, 8491843895

Offset: 1

Eric W. Weisstein, Jul 12 2003

Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph

Cf. A002218, A052443, A052445, A086216, A259862, A324234, A324240.

a(n) = A324240(n) + A324234(n). - Andrew Howroyd, Sep 04 2019

a(n) = A086216(n) - A052445(n). - Jean-François Alcover, Sep 13 2019, after Andrew Howroyd in A086216

Offset corrected by Travis Hoppe, Apr 11 2014
a(10) from the Encyclopedia of Finite Graphs (Travis Hoppe and Anna Petrone), Apr 11 2014
a(11) by Jens M. Schmidt, Feb 20 2019
a(12) added by Georg Grasegger, Jan 07 2025

A324089 Number of simple non-isomorphic n-vertex graphs of connectivity 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 121, 32960, 75032363, 345212067543

Offset: 1

Jens M. Schmidt, Feb 15 2019

Jens M. Schmidt, Data files in graph6 format

Column k=9 of A259862.

Cf. A259862, A052445.

geng $n -d8 -c | countg -G8 # Georg Grasegger, Jan 07 2025

a(13) added by Brendan McKay, Sep 01 2023

a(14) added by Georg Grasegger, Jan 07 2025

A324090 Number of simple non-isomorphic n-vertex graphs of connectivity 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 179, 113778, 814270066

Offset: 1

Jens M. Schmidt, Feb 15 2019

Jens M. Schmidt, Data files in graph6 format

Cf. A259862, A052445.

a(14) added by Brendan McKay, Sep 01 2023

A324092 Number of 7-connected simple non-isomorphic n-vertex graphs.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 5, 87, 9940, 7532629, 12213260468

Offset: 1

Jens M. Schmidt, Feb 15 2019

Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph

Cf. A259862, A086216, A259862, A324088, A324093.

a(n) = A324093(n) + A324088(n). - Andrew Howroyd, Sep 04 2019

a(13) added by Georg Grasegger, Jan 07 2025

A324093 Number of 8-connected simple non-isomorphic n-vertex graphs.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 127, 33146, 75146405, 346026751657

Offset: 1

Jens M. Schmidt, Feb 15 2019

Jens M. Schmidt, Data files in graph6 format

Cf. A259862, A086216, A259862, A324089, A324094.

a(n) = A324094(n) + A324089(n). - Andrew Howroyd, Sep 04 2019

a(13)-a(14) added by Georg Grasegger, Jan 07 2025

A324094 Number of 9-connected simple non-isomorphic n-vertex graphs.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 186, 114042, 814684114

Offset: 1

Jens M. Schmidt, Feb 15 2019

Jens M. Schmidt, Data files in graph6 format

Cf. A259862, A086216, A259862, A324090, A324095.

a(n) = A324095(n) + A324090(n). - Andrew Howroyd, Sep 04 2019

a(14) added by Georg Grasegger, Jan 07 2025

cp's OEIS Frontend

A052444 Number of simple unlabeled n-node graphs of connectivity 3.

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A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1.

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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.

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A327805 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.

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A086217 Number of 5-connected graphs on n nodes.

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A324089 Number of simple non-isomorphic n-vertex graphs of connectivity 8.

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A324090 Number of simple non-isomorphic n-vertex graphs of connectivity 9.

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A324092 Number of 7-connected simple non-isomorphic n-vertex graphs.

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A324093 Number of 8-connected simple non-isomorphic n-vertex graphs.

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A324094 Number of 9-connected simple non-isomorphic n-vertex graphs.