A000421 Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.

1, 2, 6, 20, 91, 509, 3608, 31856, 340416, 4269971, 61133757, 978098997, 17228295555, 330552900516, 6853905618223, 152626436936272, 3631575281503404, 91928898608055819, 2466448432564961852, 69907637101781318907

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more

Comments

a(n) is also the number of isomorphism classes of connected 3-regular simple graphs of order 2n with possibly loops. - Nico Van Cleemput, Jun 04 2014

There are no graphs of order 2n+1 satisfying the condition above. - Natan Arie Consigli, Dec 20 2019

Examples

			From _Natan Arie Consigli_, Dec 20 2019: (Start)
a(1) = 1: with two nodes the only viable option is the triple edged path multigraph.
a(2) = 4: with four nodes we have two cases: the tetrahedral graph and the square graph with single and double edges on opposite sides.
(End)
		

References

• A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92 [gives incorrect a(6)].
• CRC Handbook of Combinatorial Designs, 1996, p. 651 [or: 2006, table 4.40].

Links

• Jan-Peter Börnsen, Anton E. M. van de Ven, Tangent Developable Orbit Space of an Octupole, arXiv:1807.04817 [hep-th], 2018.
• G. Brinkmann, N. Van Cleemput, T. Pisanski, Generation of various classes of trivalent graphs, Theoretical Computer Science 502, 2013, pp.16-29.
• R. J. Mathar, Cubic multigraphs A000421
• Brendan McKay and others, Nauty Traces

Crossrefs

Column k=3 of A328682 (table of k-regular n-node multigraphs).

Cf. A129416, A005967 (loops allowed), A129417, A129419, A129421, A129423, A129425, A002851 (no multiedges).

Programs

nauty

for n in {1..10}; do geng -cqD3 $[2*$n] | multig -ur3; done # Sean A. Irvine, Sep 24 2015

Formula

Inverse Euler transform of A129416. - Andrew Howroyd, Mar 19 2020

Extensions

More terms from Brendan McKay, Apr 15 2007

a(13)-a(20) from Andrew Howroyd, Mar 19 2020