A002218 -id:A002218 - cp's OEIS frontend

A324235 Number of simple non-isomorphic n-vertex graphs of connectivity 6.

0, 0, 0, 0, 0, 0, 1, 4, 54, 3124, 820956, 479476753, 463431332079

Offset: 1

Jens M. Schmidt, Feb 19 2019

Jens M. Schmidt, Data files in graph6 format

Column k=6 of A259862.

geng $n -d6 -c | countg -G6 # Georg Grasegger, Jan 07 2025

a(12) added by Brendan McKay, Sep 01 2023

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

A324240 Number of simple 6-connected non-isomorphic n-vertex graphs.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 5, 59, 3211, 830896, 487009382, 475644592547

Offset: 1

Jens M. Schmidt, Feb 19 2019

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

Cf. A002218, A052443, A259862, A324092, A324235.

a(n) = A324092(n) + A324235(n). - Andrew Howroyd, Sep 04 2019

a(12)-a(13) added by Georg Grasegger, Jan 07 2025

A325447 Triangle read by rows: T(n,k) is the number of simple nonseparable (or 2-connected) graphs on n unlabeled nodes with circumference k, (n >= 3, k >= 3).

Original entry on oeis.org

1, 0, 3, 0, 2, 8, 0, 2, 6, 48, 0, 2, 9, 74, 383, 0, 2, 10, 159, 756, 6196, 0, 2, 13, 276, 2058, 14634, 177083, 0, 2, 14, 505, 4824, 48137, 384942, 9305118

Offset: 3

Andrew Howroyd, Sep 06 2019

			Triangle begins, (n >= 3, k >= 3):
  1;
  0, 3;
  0, 2,  8;
  0, 2,  6,  48;
  0, 2,  9,  74,  383;
  0, 2, 10, 159,  756,  6196;
  0, 2, 13, 276, 2058, 14634, 177083;
  0, 2, 14, 505, 4824, 48137, 384942, 9305118;
  ...

Eric Weisstein's World of Mathematics, Graph Circumference
Eric Weisstein's World of Mathematics, Hamiltonian Graph

Row sums are A002218(n >= 3).
Right diagonal is A003216.
Cf. A128042, A325455.

T(n, 4) = 2 for n > 4.

A327365 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with vertex-connectivity >= k.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 2, 1, 0, 7, 6, 3, 1, 0, 23, 21, 10, 3, 1, 0, 122, 112, 56, 17, 4, 1, 0, 888, 853, 468, 136, 25, 4, 1, 0, 11302, 11117, 7123, 2388, 384, 39, 5, 1, 0, 262322, 261080, 194066, 80890, 14480, 1051, 59, 5, 1, 0, 11730500, 11716571, 9743542, 5114079, 1211735, 102630, 3211, 87, 6, 1, 0

Offset: 0

Gus Wiseman, Sep 26 2019

A graph is covering if there are no isolated vertices.

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.

			Triangle begins:
   1
   0  0
   1  1  0
   2  2  1  0
   7  6  3  1  0
  23 21 10  3  1  0

Gus Wiseman, The graphs counted in row n = 4.

Column k = 0 is A002494.
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 non-covering version is A327805, from which this sequence differs only in the k = 0 column.

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

A331422 Triangle T(n, k) of the number of connected graphs of order n with cutting number k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 3, 0, 2, 1, 10, 0, 0, 5, 3, 2, 1, 56, 0, 0, 0, 29, 0, 13, 8, 3, 2, 1, 468, 0, 0, 0, 0, 219, 0, 0, 63, 69, 0, 16, 12, 3, 2, 1, 7123, 0, 0, 0, 0, 0, 2706, 0, 0, 0, 502, 263, 300, 0, 85, 80, 24, 16, 12, 3, 2, 1, 194066, 0, 0, 0, 0, 0, 0, 52879, 0, 0, 0, 0, 6191, 3197, 0, 2148, 861, 632, 319, 352, 132, 160, 80, 24, 21, 12, 3, 2, 1

Offset: 1

Sean A. Irvine, Jan 16 2020

The cutting number of a node v in a graph G is the number of pairs of nodes {u,w} of G such that u!=v, w!=v, and every path from u to w contains v. The cutting number of a connected graph, is the maximum cutting number of any node in the graph.

			The triangle begins:
    1;
    1;
    1, 1;
    3, 0, 2, 1;
   10, 0, 0, 5,  3,   2,  1;
   56, 0, 0, 0, 29,   0, 13, 8,  3,  2, 1;
  468, 0, 0, 0,  0, 219,  0, 0, 63, 69, 0, 16, 12, 3,  2, 1;
  ...
The length of row n is 1 + (n-1)*(n-2)/2.

Sean A. Irvine, Rows n = 1..12 flattened
Sean A. Irvine, Java program (github)
Simon Mukwembi and Senelani Dorothy Hove-Musekwa, On bounds for the cutting number of a graph, Indian J. Pure Appl. Math., 43 (2012), 637-649.

Cf. A331238 (trees), A001349 (row sums), A002218 (first column).

A340028 Number of unlabeled 2-connected graphs with n vertices rooted at a pair of noninterchangeable vertices.

Original entry on oeis.org

0, 1, 1, 7, 55, 655, 11147, 287791, 11787747, 804475261, 94875366649, 19825870580671, 7466490852631207, 5129453728126116131, 6487332587944013948099, 15213161506747424007012971, 66536415576917924594383104139, 545371527333985035460963541248785

Offset: 1

Andrew Howroyd, Jan 02 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..30

Cf. A002218, A004115, A241768, A304070, A304074, A340029.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(g=graphsSeries(n), gcr=sPoint(g)/g); x*sPoint(sSolve( sLog( gcr/(x*sv(1)) ), gcr ))}
{ my(N=15); Vec(OgfSeries(cycleIndexSeries(N)), -N) }

A340029 Number of unlabeled 2-connected graphs with n vertices rooted at a pair of indistinguishable vertices.

Original entry on oeis.org

0, 1, 1, 6, 37, 388, 6004, 148759, 5974184, 404509191, 47552739892, 9923861406343, 3735194287263442, 2565376853616300801, 3244070698095148283628, 7607050619214875184974489, 33269229977451262849539412860, 272689940536978851416633440863567

Offset: 1

Andrew Howroyd, Jan 02 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..30

Cf. A002218, A004115, A241768, A303829, A303831, A340028.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
blockGraphs(n)={my(gc=sLog(graphsSeries(n)), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)}
cycleIndexSeries(n)={sCartProd(blockGraphs(n), x^2 * symGroupCycleIndex(2) * symGroupSeries(n-2))}
{ my(N=15); Vec(OgfSeries(cycleIndexSeries(N)), -N) }

A289470 Number of strictly 2-connected graphs with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 7, 15, 39, 107, 324, 1072, 3778, 14228, 56568, 235449, 1021381, 4596328, 21383982, 102594132, 506544749, 2569520447, 13372902590, 71322154244, 389402949706

Offset: 1

Ed Pegg Jr, Jul 06 2017

Cf. A002218, A002840, A010355, A289471, A338511.

a(n) = A010355(n) - A338511(n). - Andrew Howroyd, May 03 2021

a(12)-a(13) corrected and a(14)-a(25) from Andrew Howroyd, May 03 2021

A322399 Number of non-isomorphic 2-edge-connected clutters spanning n vertices.

Original entry on oeis.org

0, 0, 2, 12, 149

Offset: 1

Gus Wiseman, Dec 06 2018

A clutter is a connected antichain of sets. It is 2-edge-connected if it cannot be disconnected by removing any single edge. Compare to blobs or 2-vertex-connected clutters (A304887).

			Non-isomorphic representatives of the a(4) = 12 clutters:
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A002218, A007718, A013922, A030019, A048143, A095983, A304887, A317634, A317635, A317672, A322335, A322394, A322396, A322397.

A337178 Number of biconnected geodetic graphs with n unlabeled vertices.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 3, 4, 3, 1, 9, 2, 4, 8, 6, 5, 13, 3, 13, 19, 11, 3, 32

Offset: 1

John Cu and Murray Elder, Jan 28 2021

A graph is geodetic if each pair of vertices is joined by a unique shortest path. A vertex v of a connected graph G is a cut vertex if G-v is disconnected. A connected graph G is biconnected if it has no cut vertices. To obtain this sequence, non-isomorphic (biconnected) graphs were generated using Brendan McKay's nauty program, then the geodetic property was checked on this output.

			For n=5 there are exactly a(5)=2 biconnected geodetic graphs: a 5-cycle and the complete graph on 5 vertices.

John Cu, Length of Embedded Circuits in Geodetic Graphs, AMSI Vacation Research Scholar project, 2021.
Brendan McKay and Adolfo Piperno, nauty and Traces. [nauty and Traces are programs for computing automorphism groups of graphs and digraphs.]
K. R. Parthasarathy and N. Srinivasan, Some general constructions of geodetic blocks, Journal of Combinatorial Theory, Series B (33) Issue 2, October 1982, Pages 121-136.
Florian Stober and Armin Weiß, Geodetic Graphs: Experiments and New Constructions, arXiv:2308.08970 [math.CO], 2023.
Wikipedia, Geodetic graph

Cf. A002218, A337179.

a(12)-a(25) from Florian Stober and Armin Weiß added by Murray Elder, Nov 14 2023

cp's OEIS Frontend

A324235 Number of simple non-isomorphic n-vertex graphs of connectivity 6.

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nauty

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A324240 Number of simple 6-connected non-isomorphic n-vertex graphs.

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A325447 Triangle read by rows: T(n,k) is the number of simple nonseparable (or 2-connected) graphs on n unlabeled nodes with circumference k, (n >= 3, k >= 3).

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A327365 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with vertex-connectivity >= k.

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A331422 Triangle T(n, k) of the number of connected graphs of order n with cutting number k >= 0.

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A340028 Number of unlabeled 2-connected graphs with n vertices rooted at a pair of noninterchangeable vertices.

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PARI

A340029 Number of unlabeled 2-connected graphs with n vertices rooted at a pair of indistinguishable vertices.

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PARI

A289470 Number of strictly 2-connected graphs with n edges.

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A322399 Number of non-isomorphic 2-edge-connected clutters spanning n vertices.

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A337178 Number of biconnected geodetic graphs with n unlabeled vertices.

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