A010355 -id:A010355 - cp's OEIS frontend

A002218 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n nodes.

0, 1, 1, 3, 10, 56, 468, 7123, 194066, 9743542, 900969091, 153620333545, 48432939150704, 28361824488394169, 30995890806033380784, 63501635429109597504951, 244852079292073376010411280, 1783160594069429925952824734641, 24603887051350945867492816663958981

Offset: 1

N. J. A. Sloane

By definition, a(n) gives the number of graphs with zero cutpoints. - Travis Hoppe, Apr 28 2014
For n > 2, a(n) is also the number of simple biconnected graphs on n nodes. - Eric W. Weisstein, Dec 07 2021
This sequence follows R. W. Robinson's definition of a nonseparable graph which includes K_2 but not the singleton graph K_1. This definition is most suited to graphical enumeration. Other authors sometimes include K_1 as a block or exclude K_2 as not 2-connected. - Andrew Howroyd, Feb 26 2023

P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 188.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..40 [terms 1..26 from R. W. Robinson]
P. Butler and R. W. Robinson, On the computer calculation of the number of nonseparable graphs, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp. [Annotated scanned copy]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
R. W. Robinson, Computer print-out of first 26 terms [Annotated scanned copy]
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]
R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 9 (1970), 327-356.
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308.
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308.
Andrés Santos, Density Expansion of the Equation of State, in A Concise Course on the Theory of Classical Liquids, Volume 923 of the series Lecture Notes in Physics, pp 33-96, 2016. DOI:10.1007/978-3-319-29668-5_3. See Reference 40.
Andrew J. Schultz and David A. Kofke, Fifth to eleventh virial coefficients of hard spheres, Phys. Rev. E 90, 023301, 4 August 2014
D. Stolee, Isomorph-free generation of 2-connected graphs with applications, arXiv preprint arXiv:1104.5261 [math.CO], 2011.
Rodrigo Stange Tessinari, Marcia Helena Moreira Paiva, Maxwell E. Monteiro, Marcelo E. V. Segatto, Anilton Garcia, George T. Kanellos, Reza Nejabati, and Dimitra Simeonidou, On the Impact of the Physical Topology on the Optical Network Performance, IEEE British and Irish Conference on Optics and Photonics (BICOP 2018), London.
Eric Weisstein's World of Mathematics, Biconnected Graph
Eric Weisstein's World of Mathematics, k-Connected Graph

Column k=0 of A325111 (for n>1).
Column sums of A339070.
Row sums of A339071.
The labeled version is A013922.
Cf. A000088 (graphs), A001349 (connected graphs), A006289, A006290, A004115 (rooted case), A010355 (by edges), A241767.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(g=graphsSeries(n), gc=sLog(g), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)}
{ my(N=12); Vec(OgfSeries(cycleIndexSeries(N)), -N) } \\ Andrew Howroyd, Dec 28 2020

More terms from Ronald C. Read. Robinson and Walsh list the first 26 terms.
a(1) changed from 0 to 1 by Eric W. Weisstein, Dec 07 2021
a(1) restored to 0 by Andrew Howroyd, Feb 26 2023

A338511 Number of unlabeled 3-connected graphs with n edges.

Original entry on oeis.org

1, 0, 1, 3, 4, 7, 22, 51, 152, 501, 1739, 6548, 26260, 110292, 483545, 2198726, 10327116, 49965520, 248481062, 1267987437, 6630660484, 35492360163, 194283212876

Offset: 6

Andrew Howroyd, Oct 31 2020

The smallest 3-connected graph is the complete graph on 4 vertices which has 6 edges.

Hugo Pfoertner, 3-connected graphs for n<=18 edges, list in PARI-readable format.
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table VI, pages 19-27.

Column sums of A339072.

Cf. A002880, A006290, A010355, A338414.

\\ It is assumed that the 3cc.gp file (from the linked zip archive) has been read before, i.e., \r [path]3cc.gp
for(k=1,#ThreeConnectedData,print1(#ThreeConnectedData[k],", "));
\\ printing of the graphs for n <= 9
print(ThreeConnectedData[6..9]) \\ Hugo Pfoertner, Dec 11 2020

a(17)-a(25) from Hugo Pfoertner using data from Robinson's tables, Nov 20 2020

a(26)-a(28) from Andrew Howroyd using data from Robinson's tables, Nov 24 2020

A339070 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 3, 3, 1, 0, 0, 0, 0, 2, 9, 4, 1, 0, 0, 0, 0, 1, 14, 20, 6, 1, 0, 0, 0, 0, 1, 12, 50, 40, 7, 1, 0, 0, 0, 0, 0, 8, 82, 161, 70, 9, 1, 0, 0, 0, 0, 0, 5, 94, 429, 433, 121, 11, 1, 0, 0, 0, 0, 0, 2, 81, 780, 1729, 1034, 189, 13, 1, 0

Offset: 1

Andrew Howroyd, Nov 23 2020

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
  1;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 0, 1, 1,  0;
  0, 0, 1, 2,  1,  0;
  0, 0, 0, 3,  3,  1,   0;
  0, 0, 0, 2,  9,  4,   1,   0;
  0, 0, 0, 1, 14, 20,   6,   1,   0;
  0, 0, 0, 1, 12, 50,  40,   7,   1,  0;
  0, 0, 0, 0,  8, 82, 161,  70,   9,  1, 0;
  0, 0, 0, 0,  5, 94, 429, 433, 121, 11, 1, 0;
  ...

Hugo Pfoertner, Table of n, a(n) for n = 1..325 (rows 1..25, first 18 rows extracted from Robinson's tables, rows 19-20 from Andrew Howroyd)
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table IV, pages 4-9.

Row sums are A010355.
Column sums are A002218.
Cf. A054923, A123534, A253186, A339071 (transpose), A339160.

T(n, n) = 1 for n >= 3.

T(n, n-1) = A253186(n-3) for n >= 3.

First row and column removed by Andrew Howroyd, Dec 05 2020

A339071 Triangle read by rows: T(n,k) is the number of unlabeled simple nonseparable (or 2-connected) graphs with n nodes and k edges (n >= 1, n-1 <= k <= n*(n-1)/2).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1, 0, 1, 4, 20, 50, 82, 94, 81, 59, 38, 20, 10, 5, 2, 1, 1, 0, 1, 6, 40, 161, 429, 780, 1076, 1197, 1114, 885, 622, 386, 215, 112, 55, 24, 11, 5, 2, 1, 1, 0, 1, 7, 70, 433, 1729, 4796

Offset: 1

Andrew Howroyd, Nov 23 2020

			Triangle T(n,k) begins:
======================================================
n/k | 0  1  2  3  4  5  6  7  8   9  10 11 12 13 14 15
----+-------------------------------------------------
  1 | 0;
  2 |    1;
  3 |       0, 1;
  4 |          0, 1, 1, 1;
  5 |             0, 1, 2, 3, 2,  1,  1;
  6 |                0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..576 (rows 1..16, extracted from Robinson's tables)
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table IV, pages 4-9.
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]

Row sums are A002218.
Column sums are A010355.
Cf. A054923, A054924, A123534, A339070 (transpose), A339072.

A010357 Number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges.

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 32, 90, 279, 942, 3468, 13777, 57747, 254671, 1170565, 5580706, 27487418, 139477796, 727458338, 3893078684, 21346838204, 119787629215, 687200870250

Offset: 1

N. J. A. Sloane

Original name: Multi-edge stars with n edges.

			From _Andrew Howroyd_, Nov 23 2020: (Start)
The a(1) = 1 graph is a single edge (K_2 = P_2).
The a(2) = 1 graph is a double edge.
The a(3) = 2 graphs are a triple edge and the triangle (K_3).
The a(4) = 3 graphs are a quadruple edge, a triangle with one double edge and the square (C_4).
(End)

George A. Baker Jr. and John M. Kincaid, The continuous-spin Ising model, g0:phi4:d field theory, and the renormalization group, J. Statist. Phys. 24 (1981), no. 3, 469-528.
Brendan McKay and Adolfo Piperno, nauty and Traces, programs for computing automorphism groups of graphs and digraphs.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 1 through a(6) = 14 unlabeled 2-connected multigraphs.

Row sums of A339160.
Cf. A050535, A076864, A010355, A010359.
A002218 counts unlabeled 2-connected graphs.
A013922 counts labeled 2-connected graphs.
A322140 is a labeled version.
Cf. A002905, A006444, A007718, A275307, A304887.

Name changed by Andrew Howroyd, Dec 05 2020
a(11)-a(20) added using geng/multig from nauty by Andrew Howroyd, Dec 05 2020
a(21)-a(23) from Sean A. Irvine, Apr 18 2024

A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 7, 16, 41, 108, 320, 1042, 3575, 13064, 49938, 197729, 805991, 3363084, 14302891, 61813285, 270805177, 1200460492, 5376709415, 24302430375, 110745093999, 508380790741

Offset: 1

Andrew Howroyd, May 04 2021

Terms may be computed using the tools geng and planarg in nauty.

Brendan McKay and Adolfo Piperno, nauty and Traces

Row sums of A343870.
Column sums of A049336(n > 1).
Cf. A002840 (3-connected), A010355, A021103, A046091, A289471, A291841.

# count graphs for the sequence by number of vertices v, sum over v afterwards
geng -C $v $n:$n | planarg -q | countg -q # Georg Grasegger, Jun 05 2023

a(21)-a(26) added by Georg Grasegger, Jun 05 2023

A339068 Number of unlabeled series-reduced 2-connected graphs with n edges.

Original entry on oeis.org

1, 0, 1, 3, 5, 9, 26, 65, 193, 632, 2173, 8049, 31690, 130773, 563849, 2525348, 11702264, 55948154, 275331956, 1392136462, 7221272289, 38380011748, 208778309856

Offset: 6

Andrew Howroyd, Nov 24 2020

Column sums of A339069.

Cf. A006289, A010355, A338511.

A010356 Triangle of single-edge stars with n edges by cyclotomic index.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 9, 2, 1, 6, 20, 14, 1, 1, 7, 40, 50, 12, 1

Offset: 3

N. J. A. Sloane

The rows appear to be the same as in A339070 but reflected and with all 0's removed. - Andrew Howroyd, Nov 25 2020

			Triangle begins:
   3 | 1;
   4 | 1;
   5 | 1, 1;
   6 | 1, 2, 1;
   7 | 1, 3, 3;
   8 | 1, 4, 9, 2;
   9 | 1, 6, 20, 14, 1;
  10 | 1, 7, 40, 50, 12, 1;
  ...

George A. Baker Jr. and John M. Kincaid, The continuous-spin Ising model, g0:phi4:d field theory and the renormalization group. J. Statist. Phys. 24 (1981), no. 3, 469-528.

Row sums give A010355.

Cf. A010358, A339070.

Offset corrected by Andrew Howroyd, Nov 25 2020

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

A360031 a(n) is the number of unlabeled 2-connected graphs with n edges containing at least one pair of nodes with resistance distance 1 when all edges are replaced by unit resistors.

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 14, 35, 111, 341, 1130, 3969, 15002, 58429, 239045, 1012241

Offset: 3

Hugo Pfoertner, Mar 11 2023

Hugo Pfoertner, Illustration of terms a(3)-a(7), 2023.

Cf. A010355, A339070, A342558, A360030.

cp's OEIS Frontend

A002218 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n nodes.

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A338511 Number of unlabeled 3-connected graphs with n edges.

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A339070 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

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A339071 Triangle read by rows: T(n,k) is the number of unlabeled simple nonseparable (or 2-connected) graphs with n nodes and k edges (n >= 1, n-1 <= k <= n*(n-1)/2).

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A010357 Number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges.

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Extensions

A343869 Number of unlabeled nonseparable (or 2-connected) planar graphs with n edges.

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Programs

nauty

Extensions

A339068 Number of unlabeled series-reduced 2-connected graphs with n edges.

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Crossrefs

A010356 Triangle of single-edge stars with n edges by cyclotomic index.

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Examples

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

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Formula

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A360031 a(n) is the number of unlabeled 2-connected graphs with n edges containing at least one pair of nodes with resistance distance 1 when all edges are replaced by unit resistors.

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