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

1,2

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

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

Table of n, a(n) for n=1..20.

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

Formula

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

Example

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)

Prog

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

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).

Sequence in context: A027321 A027315 A005965 * A009244 A104985 A210690

Adjacent sequences: A000418 A000419 A000420 * A000422 A000423 A000424

Keyword

nonn,hard,more

Author

N. J. A. Sloane

Extensions

More terms from Brendan McKay, Apr 15 2007

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

Status

approved