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

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

Author

Sean A. Irvine, Jan 16 2020

Keywords

• nonn
• tabf

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

• Sean A. Irvine, Rows n=1..27 flattened
• Frank Harary and Peter J. Slater, A linear algorithm for the cutting center of a tree, Information Processing Letters, 23 (1986), 317-319.
• Sean A. Irvine, Java program (github)
• Simon Mukwembi and Senelani Dorothy Hove-Musekwa, On bounds for the cutting number of a graph, Indian J. Pure Appl. Math., 43 (2012), 637-649.

Crossrefs

Cf. A000055 (row sums), A002887, A002888, A331237.