cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A002887 The minimum number of nodes of a tree with a cutting center of n nodes.

Original entry on oeis.org

3, 4, 7, 10, 50
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

The cutting number of a node v in a graph G is the number of pairs of nodes {u,w} of G such that u!=v, w!=v, and every path from u to w contains v. The cutting number of a connected graph (including trees as considered here), is the maximum cutting number of any node in the graph. The cutting center of a graph is the set of nodes with cutting number equal to the cutting number of the graph. - Sean A. Irvine, Jan 16 2020

References

  • Frank Harary and Phillip A. Ostrand, How cutting is a cut point?, pp. 147-150 of R. K. Guy et al., editors, Combinatorial Structures and Their Applications (Proceedings Calgary Conference Jun 1969), Gordon and Breach, NY, 1970.
  • 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).

Links

  • F. Harary and P. A. Ostrand, How cutting is a cut point?, pp. 147-150 of R. K. Guy et al., editors, Combinatorial Structures and Their Applications (Proceedings Calgary Conference Jun 1969), Gordon and Breach, NY, 1970. [Annotated scan of page 147 only.]
  • F. Harary and P. A. Ostrand, How cutting is a cut point?, pp. 147-150 of R. K. Guy et al., editors, Combinatorial Structures and Their Applications (Proceedings Calgary Conference Jun 1969), Gordon and Breach, NY, 1970. [Annotated scan of pages 148, 149 only.]
  • Frank Harary and Phillip A. Ostrand, The cutting center theorem for trees, Discrete Mathematics, 1 (1971), 7-18.

Crossrefs

Cf. A002888, A331237.

Extensions

More detailed name from R. J. Mathar, Jan 16 2020

A331237 Total cutting number of all trees of order n.

Original entry on oeis.org

0, 0, 1, 5, 15, 47, 127, 363, 978, 2778, 7624, 21566, 60584, 172221, 488978, 1398457, 4001323, 11490717, 33037548, 95195793, 274609124, 793298293, 2294114542, 6641070332, 19241453969, 55795142707, 161910611244
Offset: 1

Views

Author

Sean A. Irvine, Jan 13 2020

Keywords

Links

Crossrefs

Cf. A002887, A002888, A331236, A331238.

Formula

a(n) = Sum_{T} c(T) where the sum is over all trees with n vertices and c(T) is the cutting number of T.
a(n) = Sum_{k=0..(n-1)*(n-2)/2} A331238(n, k).

A331238 Triangle T(n, k) of the number of trees of order n with cutting number k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 7, 2, 2, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 4, 8, 4, 7, 7, 2, 3, 3, 1, 1, 1
Offset: 1

Views

Author

Sean A. Irvine, Jan 16 2020

Keywords

Comments

The cutting number of a node v in a graph G is the number of pairs of nodes {u,w} of G such that u!=v, w!=v, and every path from u to w contains v. The cutting number of a connected graph (including trees as considered here), is the maximum cutting number of any node in the graph.

Examples

			Triangle begins:
  1;
  1;
  0, 1;
  0, 0, 1, 1;
  0, 0, 0, 0, 1, 1, 1;
  0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 3, 1, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 7, 2, 2, 3, 1, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 4, 8, 4, 7, 7, 2, 3, 3, 1, 1, 1;
...
The smallest nonzero entry on each row occurs at n-2 and the maximum at (n-1)*(n-2)/2.

Links

Crossrefs

Cf. A000055 (row sums), A002887, A002888, A331237.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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