A175847 Number of cyclically 4-connected simple cubic graphs on 2n vertices.

1, 1, 2, 5, 18, 84, 607, 6100, 78824, 1195280, 20297600, 376940415, 7565248679

Offset: 2

Author

R. J. Mathar, Sep 26 2010

Keywords

• nonn
• more

Comments

The (edge-)connected simple cubic graphs counted in A002851 can be classified as 1-connected (containing bridges), 2-connected, and 3-connected. The 3-connected graphs are subdivided in the cases (i) allowing a cut of 3 edges which leaves subgraphs with cycles and (ii) cyclically 4-connected and counted here. (Computed by adding the rows with k>=4 in Brouder's Table 1.)

Each of the non-isomorphic cyclically 4-connected graphs defines a 3n-j symbol of the vector coupling coefficients in the quantum mechanics of SO(3), one 6j symbol, one 9j symbol, two 12j symbols, five 15j symbols etc.

The Yutsis graphs (A111916) are a subset of the cyclically 4-connected graphs, which admit a representation as vertex-induced binary trees.

The value a(8)=576 is found in some earlier literature (e.g., Durr et al.) - R. J. Mathar, Sep 06 2011

Examples

			On 4 vertices we have a(2)=1, the tetrahedron.
On 6 vertices we count K_4 as a(3)=1, but not the utility graph.
		

References

• A. P. Yutsis, I. B. Levinson, V. V. Vanagas, A. Sen, Mathematical apparatus of the theory of angular momentum, (1962).

Links

• G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
• Gunnar Brinkmann, Jan Goedgebeur, Jonas Hagglund and Klas Markstrom, Generation and properties of snarks, arXiv:1206.6690 [math.CO], 2012-2013.
• B. Brinkmann, J. Goedgebeur and B. D. McKay, Generation of cubic graphs, Discr. Math. Theor. Comp. Sci. 13 (2) (2011) 69-80.
• Christian Brouder and Gunnar Brinkmann, Theo Thole and the graphical methods, J. Electr. Spectr. Relat. Phen. 86 (1-3) (1997) 127-132.
• H. P. Dürr and F. Wagner, Graphical methods for the execution of the gamma or sigma-algebra in spinor theories, Nuov. Cim. 53A (1) (1968) 255.
• J.-N. Massot, E. El-Baz and J. Lafoucrière, A general graphical method for angular momentum, Rev. Mod. Phys. 39 (2) (1967) 288-305.
• M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
• Wikipedia, Table of simple cubic graphs.

Crossrefs

The labeled graphs in this class are counted by A007101. - Brendan McKay, Sep 23 2010

Extensions

Extended by Nico Van Cleemput, Jan 26 2014