A123534 -id:A123534 - cp's OEIS frontend

A013922 Number of labeled connected graphs with n nodes and 0 cutpoints (blocks or nonseparable graphs).

0, 1, 1, 10, 238, 11368, 1014888, 166537616, 50680432112, 29107809374336, 32093527159296128, 68846607723033232640, 290126947098532533378816, 2417684612523425600721132544, 40013522702538780900803893881856

Offset: 1

Stanley Selkow (sms(AT)owl.WPI.EDU)

Or, number of labeled 2-connected graphs with n nodes.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p.402.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 9.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.20(b), g(n).

Andrew Howroyd, Table of n, a(n) for n = 1..50 (terms 1..25 from R. W. Robinson)
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
Thomas Lange, Biconnected reliability, Hochschule Mittweida (FH), Fakultät Mathematik/Naturwissenschaften/Informatik, Master's Thesis, 2015.
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.
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Cf. A002218, A004115, A095983.

Row sums of triangle A123534.

seq[n_] := CoefficientList[Log[x/InverseSeries[x*D[Log[Sum[2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^n], x]]], x]*Range[0, n-2]!;
seq[16] (* Jean-François Alcover, Aug 19 2019, after Andrew Howroyd *)

seq(n)={Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))))), -n)} \\ Andrew Howroyd, Sep 26 2018

Harary and Palmer give e.g.f. in Eqn. (1.3.3) on page 10.

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.

A123542 Triangular array T(n,k) giving number of 3-connected graphs with n labeled nodes and k edges (n >= 4, ceiling(3*n/2) <= k <= n(n-1)/2).

Original entry on oeis.org

1, 15, 10, 1, 70, 492, 690, 395, 105, 15, 1, 5040, 28595, 58905, 63990, 42392, 18732, 5880, 1330, 210, 21, 1, 16800, 442680, 2485920, 6629056, 10684723, 11716068, 9409806, 5824980, 2872317, 1147576, 373156, 98112, 20475, 3276

Offset: 4

N. J. A. Sloane, Nov 13 2006

			Triangle begins:
n = 4
k = 6 : 1
Total( 4) = 1
n = 5
k = 8 : 15
k = 9 : 10
k = 10 : 1
Total( 5) = 26
n = 6
k = 9 : 70
k = 10 : 492
k = 11 : 690
k = 12 : 395
k = 13 : 105
k = 14 : 15
k = 15 : 1
Total( 6) = 1768
n = 7
k = 11 : 5040
k = 12 : 28595
k = 13 : 58905
k = 14 : 63990
k = 15 : 42392
k = 16 : 18732
k = 17 : 5880
k = 18 : 1330
k = 19 : 210
k = 20 : 21
k = 21 : 1
Total( 7) = 225096

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

R. W. Robinson, Rows 4 through 15, flattened (row 15 is incomplete).
T. R. S. Walsh, Counting labeled three-connected and homeomorphically irreducible two-connected graphs, J. Combin. Theory Ser. B 32 (1982), no. 1, 1-11, Table 1.

Row sums give A005644. Cf. A123527, A123534.

A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 0, 1, 3, 18, 131, 1180, 12570, 154535, 2151439, 33431046, 573197723, 10743619285, 218447494812, 4787255999220, 112454930390211, 2818138438707516, 75031660452368001, 2114705500316025737, 62890323682634277951, 1967901134191778583146, 64623905086814216468839

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, The a(6) = 131 labeled 2-connected graphs with 6 edges.

Cf. A002218, A013922, A123534, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

a(n) = Sum_{i=3..n} A123534(i, n). - Andrew Howroyd, Nov 30 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018

cp's OEIS Frontend

A013922 Number of labeled connected graphs with n nodes and 0 cutpoints (blocks or nonseparable graphs).

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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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A123542 Triangular array T(n,k) giving number of 3-connected graphs with n labeled nodes and k edges (n >= 4, ceiling(3*n/2) <= k <= n(n-1)/2).

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A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

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