A002935 -id:A002935 - cp's OEIS frontend

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).

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

A360879 Number of unlabeled nonseparable (or 2-connected) loopless multigraphs with circuit rank n and degree >= 3 at each node.

Original entry on oeis.org

0, 1, 4, 17, 118, 1198, 17133, 311757, 6803203

Offset: 1

Andrew Howroyd, Feb 25 2023

Martin Dowd, Some results on reconstructibility of colored graphs, IJPAM, 95 (2014), 309-321. Gives the sequence up to a(8) in Figure 1.
Martin Dowd, The source code of an implementation of the algorithm from the paper and data files containing the output graphs, 2024.
Zhizheng Ye, Xuewen Huang, Chuanyu Wu, Xianglei Xue and Liang Sun, Synthesis of contracted graph for planar nonfractionated simple-jointed kinematic chain based on similarity information, Mechanism and Machine Theory, 181 (2023), 105227. Gives the sequence in Table 3 with an erroneous term a(8) = 311737.

Diagonal sums of A046752.

Cf. A002935, A360864.

a(n) = Sum_{k=2..2*n-2} A046752(n + k - 1, k).

a(8) from the paper by Martin Dowd (2014) and a(9) from Martin Dowd (personal communication) added by Andrey Zabolotskiy, Feb 21 2024

A360871 Number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and degree >= 3 at each node, loops allowed.

Original entry on oeis.org

0, 0, 2, 4, 9, 20, 44, 113, 329, 1044, 3622, 13544, 53596, 223084, 969158

Offset: 1

Andrew Howroyd, Feb 25 2023

A single-edge is considered to be nonseparable here.

			The a(3) = 2 multigraphs are:
  - a triple edge;
  - a single edge with a loop at each vertex.

Row sums of A360870.

Cf. A002935, A360863, A360867.

A360882 Number of unlabeled connected multigraphs with n edges, no cut-points and degree >= 3 at each node, loops allowed.

Original entry on oeis.org

0, 1, 3, 5, 10, 21, 45, 114, 330, 1045, 3623, 13545, 53597, 223085, 969159

Offset: 1

Andrew Howroyd, Feb 27 2023

			The a(3) = 3 multigraphs are:
  - a single vertex with 3 loops;
  - a triple edge;
  - a single edge with a loop at each vertex.

Cf. A002935, A360863, A360867, A360871.

Row sums of A360870.

a(n) = A360871(n) + 1 for n > 1.

cp's OEIS Frontend

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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A360879 Number of unlabeled nonseparable (or 2-connected) loopless multigraphs with circuit rank n and degree >= 3 at each node.

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Extensions

A360871 Number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and degree >= 3 at each node, loops allowed.

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A360882 Number of unlabeled connected multigraphs with n edges, no cut-points and degree >= 3 at each node, loops allowed.

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