A166974 Number of single-component graphs where the product of the valences of the nodes is n.

1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 2, 2, 8, 1, 6, 1, 6, 2, 2, 1, 16, 2, 2, 4, 6, 1, 8, 1, 16, 2, 2, 2, 25, 1, 2, 2, 16, 1, 8, 1, 6, 6, 2, 1, 46, 2, 6, 2, 6, 1, 18, 2, 16, 2, 2, 1, 36, 1, 2, 6, 40, 2, 8, 1, 6, 2, 8, 1, 84, 1, 2, 6, 6, 2, 8, 1, 49, 12, 2, 1, 36, 2, 2, 2, 16, 1, 38, 2, 6, 2, 2, 2, 137

Offset: 0

Author

Franklin T. Adams-Watters, Oct 26 2009

Keywords

• nice
• nonn

Comments

A single-component graph is any nonempty connected graph. If the empty graph was allowed, a(1) would be 2 instead of 1.

The sequence can be computed for n > 1 by looking at the graph that results when all valence 1 nodes are removed. This will be a connected graph, and labeling each node with its original valence, the product of the labels will be the original product. Each node must be labeled with at least its valence, and at least 2. Each such labeling, up to graph equivalence, uniquely defines the original graph, so we need only count the labelings for connected graphs with up to BigOmega(n) nodes.

Note, in particular, that a(n) = 1 for any prime, and 2 for any semiprime.

This product for the complete graph on n points is (n-1)^n. For the complete bipartite graph with n and m points in the parts the product is n^m*m^n. For the cyclic graph with n nodes it is 2^n.

Links

• Andrew Weimholt, Table of n, a(n) for n = 0..255

Crossrefs

Cf. A000079, A001222 (BigOmega), A001349, A002905, A007778, A062275.

Extensions

Corrected and extended by Andrew Weimholt, Oct 26 2009