A052445 -id:A052445 - cp's OEIS frontend

A324088 Number of simple non-isomorphic n-vertex graphs of connectivity 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 4, 81, 9813, 7499483, 12138114063

Offset: 1

Jens M. Schmidt, Feb 15 2019

Jens M. Schmidt, Data files in graph6 format

Column k=7 of A259862.

Cf. A259862, A052445.

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

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).

Original entry on oeis.org

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.

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

A086216 Number of 4-connected unlabeled n-node graphs.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 25, 384, 14480, 1211735, 184649399, 47952362294

Offset: 1

Eric W. Weisstein, Jul 12 2003

The definition means that the connectivity is 4 or more.

			There are 4 different 4-connected graphs on 6 vertices. - _Dylan Thurston_, Jun 18 2009

Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, [math.CO], 2014.
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, Discr. Appl. Math. 201 (2016) 172-181.
Eric Weisstein's World of Mathematics, k-Connected Graph

See A052445 for exactly-4-connected graphs.
See A086217 for 5-connected graphs.
See A259862 for further information.

a(n) = A086217(n) + A052445(n). - Andrew Howroyd, Sep 04 2019

Offset corrected by Dylan Thurston, Jun 18 2009
a(10) from the Encyclopedia of Finite Graphs (Travis Hoppe and Anna Petrone), Apr 11 2014
Minor edits by N. J. A. Sloane, Jul 08 2015 at the suggestion of Brendan McKay.
a(12) added by Georg Grasegger, Jan 07 2025

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

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

A324091 Number of simple non-isomorphic n-vertex graphs of connectivity 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 257, 413672, 9506810555

Offset: 1

Jens M. Schmidt, Feb 15 2019

Jens M. Schmidt, Data files in graph6 format

Column k=10 of A259862.

Cf. A259862, A052445.

geng $n -d10 -c | countg -G10 # Georg Grasegger, Jan 07 2025

a(15) added by Georg Grasegger, Jan 07 2025

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A324088 Number of simple non-isomorphic n-vertex graphs of connectivity 7.

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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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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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A086216 Number of 4-connected unlabeled n-node graphs.

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

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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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A324091 Number of simple non-isomorphic n-vertex graphs of connectivity 10.

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