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A000421 Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n. 16
1, 2, 6, 20, 91, 509, 3608, 31856, 340416, 4269971, 61133757, 978098997, 17228295555, 330552900516, 6853905618223, 152626436936272, 3631575281503404, 91928898608055819, 2466448432564961852, 69907637101781318907 (list; graph; refs; listen; history; text; internal format)



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


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].


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


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


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.



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


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




N. J. A. Sloane


More terms from Brendan McKay, Apr 15 2007

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



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Last modified March 31 15:21 EDT 2023. Contains 361668 sequences. (Running on oeis4.)