A324088 -id:A324088 - 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

A001072 Number of minimally 2-edge-connected non-isomorphic graphs with n nodes.

Original entry on oeis.org

1, 1, 3, 4, 11, 23, 63, 159, 459, 1331, 4083, 12750

Offset: 3

Robert W. Robinson

Calculated by Sridar K. Pootheri.

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version d0efd85.
Jens M. Schmidt, Data files in graph6 format

Cf. A324410.

a(11) and a(12) added using tinygraph by Falk Hüffner, Jan 20 2016

a(13) and a(14) added by Jens M. Schmidt, Feb 27 2019

A052443 Number of simple unlabeled n-node graphs of connectivity 2.

Original entry on oeis.org

0, 0, 1, 2, 7, 39, 332, 4735, 113176, 4629463, 327695586, 40525166511, 8850388574939, 3453378695335727, 2435485662537561705, 3137225298932374490227, 7448146273273417700880931, 32837456713651735794742705141, 270528237651574516777595556494978, 4186091025846007046878947026003803389

Offset: 1

Eric W. Weisstein

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..23
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph.
Gus Wiseman, The a(5) = 7 graphs with vertex-connectivity 2.

Column k=2 of A259862.
Cf. A000719, A002218, A006290, A052442, A052444, A052445.
The labeled version is A327198.
2-vertex-connected graphs are A013922.
Cf. A322387, A327051, A327101, A327334.

A002218 = Cases[Import["https://oeis.org/A002218/b002218.txt", "Table"], {, }][[All, 2]];
A006290 = Cases[Import["https://oeis.org/A006290/b006290.txt", "Table"], {, }][[All, 2]];
a[1] = 0; a[2] = 0; a[3] = 1;
a[n_] := A002218[[n]] - A006290[[n-3]];
Array[a, 23] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

a(n) = A002218(n) - A006290(n) for n > 2. - Andrew Howroyd, Sep 04 2019

Name clarified and a(8)-a(11) by Jens M. Schmidt, Feb 18 2019
a(2)-a(3) corrected by Andrew Howroyd, Aug 28 2019
a(12)-a(20) from Andrew Howroyd, Sep 04 2019

A052442 Number of simple unlabeled n-node graphs of connectivity 1.

Original entry on oeis.org

0, 1, 1, 3, 11, 56, 385, 3994, 67014, 1973029, 105731474, 10439496931, 1902968718515, 641662974453892, 401490336727861176, 467924684115578671326, 1019752390010650509117288, 4171131179469162937375841939, 32134378048921787829834095722663, 467778894124037894839737804918978194

Offset: 1

Eric W. Weisstein

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..26
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph.
Eric Weisstein's World of Mathematics, Biconnected Graph

Column k=1 of A259862.

Cf. A000719, A052443, A052444, A052445.

A001349 = Cases[Import["https://oeis.org/A001349/b001349.txt", "Table"], {, }][[All, 2]];
A002218 = Cases[Import["https://oeis.org/A002218/b002218.txt", "Table"], {, }][[All, 2]];
a[1] = 0; a[2] = 1;
a[n_] := A001349[[n+1]] - A002218[[n]];
Array[a, 26] (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

a(n) = A001349(n) - A002218(n) for n > 2. - Andrew Howroyd, Sep 04 2019

Terms a(8)-a(11) by Jens M. Schmidt, Feb 18 2019
a(1)-a(2) corrected by Andrew Howroyd, Aug 28 2019
a(12)-a(20) from Andrew Howroyd, Sep 04 2019

A052445 Number of simple unlabeled n-node graphs of connectivity 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 21, 345, 13429, 1109105, 162318088, 39460518399

Offset: 1

Eric W. Weisstein

			The a(6) = 3 exactly-4-connected 6-node graphs are the complete graph K_6 with 1, 2, or 3 non-adjacent edges removed.

Brendan McKay, confusion over k-connected graphs, posting to Sequence Fans Mailing List, Jul 08 2015.
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph
Eric Weisstein's World of Mathematics, Vertex Connectivity

Cf. A052442, A052443, A052444, A086216, A086217, A259862.

a(n) = A086216(n) - A086217(n). - Andrey Zabolotskiy, Nov 20 2017

Partially edited by N. J. A. Sloane, Jul 08 2015 at the suggestion of Brendan McKay
a(8)-a(11) copied from A259862 by Andrey Zabolotskiy, Nov 20 2017
a(4)-a(5) corrected by Andrew Howroyd, Aug 28 2019
a(12) from Sean A. Irvine, Dec 12 2021

A324417 Number of minimally 10-edge-connected non-isomorphic n-vertex graphs.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 25, 2498

Offset: 1

Jens M. Schmidt, Mar 03 2019

Jens M. Schmidt, Data files in graph6 format

Cf. A001072, A324410.

A324418 Number of minimally 4-connected non-isomorphic n-vertex graphs.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 13, 83, 704, 8738, 136391

Offset: 1

Jens M. Schmidt, Mar 03 2019

Jens M. Schmidt, Data files in graph6 format

Cf. A003317, A324424.

A241704 Number of simple unlabeled n-node graphs of edge-connectivity 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 41, 1095, 104829, 21981199, 8077770931

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=5 of A263296.

Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A052448, A241703, A241705.

a(11)-a(12) by Jens M. Schmidt, Feb 18 2019

A241705 Number of simple unlabeled n-node graphs of edge-connectivity 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 65, 3441, 857365, 487560158, 466534106494

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Jens M. Schmidt, Data files in graph6 format
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=6 of A263296.

Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A052448, A241703, A241704.

a(11)-a(13) by Jens M. Schmidt, Feb 20 2019

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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A001072 Number of minimally 2-edge-connected non-isomorphic graphs with n nodes.

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A052443 Number of simple unlabeled n-node graphs of connectivity 2.

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A052442 Number of simple unlabeled n-node graphs of connectivity 1.

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A052445 Number of simple unlabeled n-node graphs of connectivity 4.

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A324417 Number of minimally 10-edge-connected non-isomorphic n-vertex graphs.

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A324418 Number of minimally 4-connected non-isomorphic n-vertex graphs.

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A241704 Number of simple unlabeled n-node graphs of edge-connectivity 5.

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A241705 Number of simple unlabeled n-node graphs of edge-connectivity 6.

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