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A002851 Number of unlabeled trivalent (or cubic) connected graphs with 2n nodes.
(Formerly M1521 N0595)
1, 0, 1, 2, 5, 19, 85, 509, 4060, 41301, 510489, 7319447, 117940535, 2094480864, 40497138011, 845480228069, 18941522184590, 453090162062723, 11523392072541432, 310467244165539782, 8832736318937756165 (list; graph; refs; listen; history; text; internal format)



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)


Table of n, a(n) for n=0..20.

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, 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, 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, Adrian Del Maestro, Moments of the inverse participation ratio for the Laplacian on finite regular graphs, arXiv:1506.02048 [math-ph], 2015.

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, G. Tinhofer, Algebraic Combinatorics (1997)

Denis S. Krotov, 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


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.


Cf. A004109 (labeled connected cubic), A321304 (rooted connected cubic), A321305 (signed connected cubic), A000421 (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)

Sequence in context: A286886 A058132 A286071 * A354621 A324618 A326563

Adjacent sequences:  A002848 A002849 A002850 * A002852 A002853 A002854




N. J. A. Sloane


More terms from R. C. Read (rcread(AT)math.uwaterloo.ca)



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