A241770 -id:A241770 - cp's OEIS frontend

A241768 Number of simple connected graphs with n nodes and exactly 2 articulation points (cutpoints).

0, 0, 0, 1, 3, 17, 101, 890, 11468, 239728

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=2 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A241767 Number of simple connected graphs with n nodes and exactly 1 articulation point (cutpoints).

Original entry on oeis.org

0, 0, 1, 2, 7, 33, 244, 2792, 52448, 1690206, 96288815, 9873721048, 1841360945834, 629414405238720, 397024508142598996, 464923623652122023478, 1016016289424631486429082, 4162473006943138723685574978, 32096861904411547975392065322659

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

nonn

Terms may be computed from A004115. See formula. There is an obvious bijection between a connected graph with 1 articulation point and a multiset of at least two rooted nonseparable graphs joined at the root node. - Andrew Howroyd, Nov 24 2020

Andrew Howroyd, Table of n, a(n) for n = 1..26
Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=1 of A325111.
Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.
Cf. A004115 (rooted and without articulation points).

G.f.: x/(Product_{k>=1} (1 - x^k)^A004115(k+1)) - x - Sum_{k>=1} A004115(k)*x^k. - Andrew Howroyd, Nov 24 2020

Terms a(11) and beyond from Andrew Howroyd, Nov 24 2020

A325111 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k articulation vertices, (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 3, 2, 1, 0, 0, 10, 7, 3, 1, 0, 0, 56, 33, 17, 5, 1, 0, 0, 468, 244, 101, 32, 7, 1, 0, 0, 7123, 2792, 890, 242, 60, 9, 1, 0, 0, 194066, 52448, 11468, 2461, 527, 97, 12, 1, 0, 0, 9743542, 1690206, 239728, 35839, 6056, 1029, 155, 15, 1, 0, 0

Offset: 0

Andrew Howroyd, Sep 05 2019

Articulation vertices are also called cutpoints. These are vertices that when removed increase the component count of the graph.

			Triangle begins:
     1;
     1     0;
     1,    0,   0;
     1,    1,   0,   0;
     3,    2,   1,   0,  0;
    10,    7,   3,   1,  0, 0;
    56,   33,  17,   5,  1, 0, 0;
   468,  244, 101,  32,  7, 1, 0, 0;
  7123, 2792, 890, 242, 60, 9, 1, 0, 0;
  ...

Eric Weisstein's World of Mathematics, Articulation Vertex

Columns k=0..5 are A002218(n>1), A241767, A241768, A241769, A241770, A241771.
Row sums are A001349.
Cf. A327077, A370064 (labeled version).

Diagonal for k = n inserted by Andrew Howroyd, Feb 25 2024

A241769 Number of simple connected graphs with n nodes and exactly 3 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 32, 242, 2461, 35839

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=3 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A241771 Number of simple connected graphs with n nodes and exactly 5 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 9, 97, 1029

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=5 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

cp's OEIS Frontend

A241768 Number of simple connected graphs with n nodes and exactly 2 articulation points (cutpoints).

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A241767 Number of simple connected graphs with n nodes and exactly 1 articulation point (cutpoints).

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A325111 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k articulation vertices, (0 <= k <= n).

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A241769 Number of simple connected graphs with n nodes and exactly 3 articulation points (cutpoints).

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A241771 Number of simple connected graphs with n nodes and exactly 5 articulation points (cutpoints).

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