A339070 -id:A339070 - 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

A010355 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 7, 16, 42, 111, 331, 1094, 3829, 14380, 57069, 237188, 1027929, 4622588, 21494274, 103077677, 508743475, 2579847563, 13422868110, 71570635306, 390670937143

Offset: 1

N. J. A. Sloane

Original name: Single-edge stars with n edges.

			From _Andrew Howroyd_, Nov 23 2020: (Start)
The a(1) = 1 graph is the single edge (K_2 = P_2).
The a(3) = 1 graph is the triangle (K_3).
The a(4) = 1 graph is the square (C_4).
The a(5) = 2 graphs are the cycle C_5 and a cycle of 4 nodes with one diagonal added.
(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, Practical Graph Isomorphism, II, Journal of Symbolic Computation, 60 (2014), pp. 94-112.
Brendan McKay and Adolfo Piperno, nauty and Traces, programs for computing automorphism groups of graphs and digraphs.
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table IV, pages 4-9.
M. F. Sykes, D. S. McKenzie, and B. R. Heap, Specific heat of a three-dimensional Ising ferromagnet above the Curie temperature. III. The star cluster expansion, J. Phys. A: Math. Nucl. Gen., 7 (1974), 1576.

Row sums of A339070 and A010356.
Column sums of A339071.
Cf. A002218, A010357, A322139, A338511.

a(11)-a(12) from Andrey Zabolotskiy, Oct 03 2017
Name changed by Andrew Howroyd, Nov 23 2020
a(13)-a(18) added using data from Robinson's tables by Andrew Howroyd, Nov 23 2020
a(19)-a(22) from Hugo Pfoertner using program geng from nauty, Dec 04 2020
a(23)-a(24) from Hugo Pfoertner, Dec 07 2020
a(25) from Hugo Pfoertner, Jan 04 2021

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.

A339160 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 3, 6, 3, 1, 0, 1, 4, 11, 11, 4, 1, 0, 1, 5, 22, 33, 23, 5, 1, 0, 1, 7, 38, 89, 96, 40, 7, 1, 0, 1, 8, 63, 212, 345, 234, 70, 8, 1, 0, 1, 10, 98, 463, 1083, 1146, 546, 110, 10, 1, 0, 1, 12, 151, 943, 3068, 4739, 3505, 1169, 176, 12, 1, 0

Offset: 1

Andrew Howroyd, Dec 05 2020

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
  1;
  1,  0;
  1,  1,   0;
  1,  1,   1,   0;
  1,  2,   2,   1,    0;
  1,  3,   6,   3,    1,    0;
  1,  4,  11,  11,    4,    1,    0;
  1,  5,  22,  33,   23,    5,    1,    0;
  1,  7,  38,  89,   96,   40,    7,    1,   0;
  1,  8,  63, 212,  345,  234,   70,    8,   1,  0;
  1, 10,  98, 463, 1083, 1146,  546,  110,  10,  1, 0;
  1, 12, 151, 943, 3068, 4739, 3505, 1169, 176, 12, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210 (rows 1..20)

Column k=3 is A001399(n-3).
Row sums are A010357.
Cf. A046752, A191646, A339070.

T(n,2) = T(n,n) = 1.

A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar 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, 13, 20, 6, 1, 0, 0, 0, 0, 0, 11, 49, 40, 7, 1, 0, 0, 0, 0, 0, 5, 77, 158, 70, 9, 1, 0, 0, 0, 0, 0, 2, 75, 406, 426, 121, 11, 1, 0, 0, 0, 0, 0, 0, 47, 662, 1645, 1018, 189, 13, 1, 0

Offset: 1

Andrew Howroyd, May 04 2021

			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, 13, 20,   6,   1,   0;
  0, 0, 0, 0, 11, 49,  40,   7,   1,  0;
  0, 0, 0, 0,  5, 77, 158,  70,   9,  1, 0;
  0, 0, 0, 0,  2, 75, 406, 426, 121, 11, 1, 0;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..351 (26 rows) (first 210 terms (20 rows) from Andrew Howroyd)

Row sums are A343869.
Column sums are A021103.
Cf. A049334, A049336 (transpose), A049337, A253186, A339070.

geng -C $k $n:$n | planarg -q | countg -q # Georg Grasegger, Jun 05 2023

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

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

A360880 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and k nodes, loops allowed, n >= 1, 2 <= k <= n + 1.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 6, 2, 1, 0, 9, 6, 3, 1, 0, 12, 14, 13, 4, 1, 0, 16, 28, 39, 22, 5, 1, 0, 20, 52, 112, 98, 39, 6, 1, 0, 25, 93, 281, 383, 236, 63, 8, 1, 0, 30, 152, 655, 1304, 1220, 515, 102, 9, 1, 0, 36, 242, 1408, 3980, 5418, 3512, 1077, 153, 11, 1, 0

Offset: 1

Andrew Howroyd, Feb 25 2023

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
   1;
   2,   0;
   4,   1,   0;
   6,   2,   1,    0;
   9,   6,   3,    1,    0;
  12,  14,  13,    4,    1,   0;
  16,  28,  39,   22,    5,   1,   0;
  20,  52, 112,   98,   39,   6,   1, 0;
  25,  93, 281,  383,  236,  63,   8, 1, 0;
  30, 152, 655, 1304, 1220, 515, 102, 9, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..190 (rows 1..19)

Columns k=2..3 are A002620(n+1), A050531(n-3).
Row sums are A360881.
Cf. A290428, A322115, A339070, A339160, A360870.

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

cp's OEIS Frontend

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

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A010355 Number of unlabeled nonseparable (or 2-connected) graphs (or blocks) with n edges.

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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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A339160 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

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A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

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nauty

Formula

A360880 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and k nodes, loops allowed, n >= 1, 2 <= k <= n + 1.

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

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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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