A339160 -id:A339160 - cp's OEIS frontend

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).

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

A046752 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges on k nodes and degree >= 3 at each node, n >= 3, 2 <= k <= floor(2*n/3).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 1, 4, 13, 4, 1, 6, 26, 24, 5, 1, 7, 47, 84, 38, 1, 9, 78, 233, 216, 23, 1, 11, 126, 557, 914, 314, 16, 1, 13, 188, 1193, 3077, 2270, 325, 1, 15, 276, 2355, 8915, 11592, 4015, 162, 1, 18, 391, 4370, 23008, 47079, 31443, 4495, 66

Offset: 3

N. J. A. Sloane

Original name: Triangle of number of homeomorphically irreducible stars with n edges and m nodes.

			Triangle begins:
  1;
  1;
  1,  1;
  1,  2,   2;
  1,  3,   5;
  1,  4,  13,    4;
  1,  6,  26,   24,    5;
  1,  7,  47,   84,   38;
  1,  9,  78,  233,  216,    23;
  1, 11, 126,  557,  914,   314,   16;
  1, 13, 188, 1193, 3077,  2270,  325;
  1, 15, 276, 2355, 8915, 11592, 4015, 162;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..93 (rows 3..18)
B. R. Heap, The enumeration of homeomorphically irreducible star graphs, J. Math. Phys., 7 (1966), 1582-1587.
Sean A. Irvine, Java program (github).

Row sums are A002935.
Diagonal sums are A360879.
Cf. A339160, A360866, A360870.

More terms from Sean A. Irvine, May 16 2020

Name edited and offset corrected by Andrew Howroyd, Feb 25 2023

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

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.

A010358 Triangle of multi-edge stars with n edges by cyclotomic index.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 4, 11, 11, 4, 1, 1, 5, 23, 33, 22, 5, 1, 1, 7, 40, 96, 89, 38, 7, 1, 1, 8, 70, 234, 345, 212, 63, 8, 1, 1, 10, 110, 546, 1146, 1083, 463, 98, 10, 1, 1, 12, 176, 1169, 3505, 4739, 3068, 943, 151, 12, 1, 1, 14, 259, 2365, 9650, 18453, 17082, 7884, 1806, 218, 14, 1

Offset: 2

N. J. A. Sloane

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.

Cf. A339160 (rows reversed).

Offset corrected, rows 11-13 added from A339160 by Andrey Zabolotskiy, Feb 17 2021

cp's OEIS Frontend

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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A046752 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges on k nodes and degree >= 3 at each node, n >= 3, 2 <= k <= floor(2*n/3).

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

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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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Examples

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A010358 Triangle of multi-edge stars with n edges by cyclotomic index.

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