A002851 Number of unlabeled trivalent (or cubic) connected simple graphs with 2n nodes.

1, 0, 1, 2, 5, 19, 85, 509, 4060, 41301, 510489, 7319447, 117940535, 2094480864, 40497138011, 845480228069, 18941522184590, 453090162062723, 11523392072541432, 310467244165539782, 8832736318937756165

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Examples

			G.f. = 1 + x^2 + 2*x^3 + 5*x^4 + 19*x^5 + 85*x^6 + 509*x^7 + 4060*x^8 + 41302*x^9 + 510489*x^10 + 7319447*x^11 + ...
a(0) = 1 because the null graph (with no vertices) is vacuously 3-regular.
a(1) = 0 because there are no simple connected cubic graphs with 2 nodes.
a(2) = 1 because the tetrahedron is the only cubic graph with 4 nodes.
a(3) = 2 because there are two simple cubic graphs with 6 nodes: the bipartite graph K_{3,3} and the triangular prism graph.
		

References

• CRC Handbook of Combinatorial Designs, 1996, p. 647.
• F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 195.
• R. C. Read, Some applications of computers in graph theory, in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, pp. 417-444.
• R. C. Read and G. F. Royle, Chromatic roots of families of graphs, pp. 1009-1029 of Y. Alavi et al., eds., Graph Theory, Combinatorics and Applications. Wiley, NY, 2 vols., 1991.
• R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence)

Links

• Peter Adams, Ryan C. Bunge, Roger B. Eggleton, Saad I. El-Zanati, Uğur Odabaşi, and Wannasiri Wannasit, Decompositions of complete graphs and complete bipartite graphs into bipartite cubic graphs of order at most 12, Bull. Inst. Combinatorics and Applications (2021) Vol. 92, 50-61.
• G. Brinkmann, J. Goedgebeur and B. D. McKay, Generation of cubic graphs, Discr. Math. Theor. Comp. Sci. 13 (2) (2011) 69-80
• G. Brinkmann, J. Goedgebeur, and N. Van Cleemput, The history of the generation of cubic graphs, Int. J. Chem. Modeling 5 (2-3) (2013) 67-89
• F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
• F. C. Bussemaker, S. Cobeljic, D. M. Cvetkovic, and J. J. Seidel, Cubic graphs on <= 14 vertices J. Combinatorial Theory Ser. B 23(1977), no. 2-3, 234--235. MR0485524 (58 #5354).
• Timothy B. P. Clark and Adrian Del Maestro, Moments of the inverse participation ratio for the Laplacian on finite regular graphs, arXiv:1506.02048 [math-ph], 2015.
• Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 1.
• H. Gropp, Enumeration of regular graphs 100 years ago, Discrete Math., 101 (1992), 73-85.
• House of Graphs, Cubic graphs
• Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
• M. Klin, M. Rücker, Ch. Rücker and G. Tinhofer, Algebraic Combinatorics [broken link]
• M. Klin, M. Rücker, Ch. Rücker, and G. Tinhofer, Algebraic Combinatorics (1997)
• Denis S. Krotov and Konstantin V. Vorob'ev, On unbalanced Boolean functions attaining the bound 2n/3-1 on the correlation immunity, arXiv:1812.02166 [math.CO], 2018.
• R. J. Mathar/Wikipedia, Table of simple cubic graphs [From _N. J. A. Sloane_, Feb 28 2012]
• M. Meringer, Tables of Regular Graphs
• R. W. Robinson and N. C. Wormald, Numbers of cubic graphs. J. Graph Theory 7 (1983), no. 4, 463-467.
• Sage, Common Graphs (Graph Generators)
• J. J. Seidel, R. R. Korfhage, & N. J. A. Sloane, Correspondence 1975
• H. M. Sultan, Separating pants decompositions in the pants complex.
• H. M. Sultan, Net of Pants Decompositions Containing a non-trivial Separating Curve in the Pants Complex, arXiv:1106.1472 [math.GT], 2011.
• Eric Weisstein's World of Mathematics, Connected Graph
• Eric Weisstein's World of Mathematics, Cubic Graph

Crossrefs

Cf. A004109 (labeled connected cubic), A361407 (rooted connected cubic), A321305 (signed connected cubic), A000421 (connected cubic loopless multigraphs), A005967 (connected cubic multigraphs), A275744 (multisets).

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)

3-regular simple graphs: this sequence (connected), A165653 (disconnected), A005638 (not necessarily connected), A005964 (planar).

Connected regular graphs A005177 (any degree), A068934 (triangular array), specified degree k: this sequence (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).

Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: this sequence (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7). (End)

Extensions

More terms from Ronald C. Read