A004109 -id:A004109 - cp's OEIS frontend

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

N. J. A. Sloane

			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.

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)

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

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)

More terms from Ronald C. Read

A002829 Number of trivalent (or cubic) labeled graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 70, 19355, 11180820, 11555272575, 19506631814670, 50262958713792825, 187747837889699887800, 976273961160363172131825, 6840300875426184026353242750, 62870315446244013091262178375075, 741227949070136911068308523257857500

Offset: 0

N. J. A. Sloane

nonn

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 411.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
R. W. Robinson, Computer print-out, no date. Gives first 30 terms.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..160 (first 31 terms from R. W. Robinson)
F. Chyzak, M. Mishna and B. Salvy, Effective D-Finite Symmetric Functions, FPSAC '02 (2002) # 19.12, see Maple output prior to the References.
Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 9.
Oleg Evnin and Weerawit Horinouchi, A Gaussian integral that counts regular graphs, arXiv:2403.04242 [cond-mat.stat-mech], 2024. See p. 12.
I. P. Goulden and D. M. Jackson, Labelled graphs with small vertex degrees and P-recursiveness, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60-66. MR0819706 (87k:05093)
I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)
R. W. Robinson, Cubic labeled graphs, computer print-out, n.d.
N. C. Wormald, Enumeration of labelled graphs II: cubic graphs with a given connectivity, J. Lond Math Soc s2-20 (1979) 1-7, eq. (2.6).

A diagonal of A059441. Cf. A005814.

See A004109 for connected graphs of this type.

From R. J. Mathar, Oct 31 2010: (Start)
A002829aux := proc(i) local a,j,k ; a := 0 ; for j from 0 to i do for k from 0 to 2*(i-j) do a := a+(-1)^(j+k)/j!*doublefactorial(2*i+2*k-1)/3^k/k!/(2*i-2*j-k)! ; end do: end do: a*3^i/2^i ; end proc:
A002829 := proc(n) (2*n)!/6^n*add( A002829aux(i)/(n-i)!,i=0..n) ; end proc: seq(A002829(n),n=0..6) ; (End)
egf := hypergeom([1/6, 5/6],[],12*x/(x^2+8*x+4)^(3/2)) * exp(-ln(1/4*x^2+2*x+1)/4 - x/3 + (x^2+8*x+4)^(3/2)/(24*x) - 1/(3*x) - x^2/24 - 1):
ser := convert(series(egf,x=0,30),polynom):
seq(coeff(ser,x,i) * (2*i)!, i=0..degree(ser)); # Mark van Hoeij, Nov 07 2011

Flatten[{1,RecurrenceTable[{2 (-3+n) (-2+n) (-1+n) (-7+2 n) (-5+2 n) (-3+2 n) (-1+2 n) (-4+3 n) (-1+3 n) a[-4+n]-2 (-2+n) (-1+n) (-5+2 n) (-3+2 n) (-1+2 n) (-1+3 n) (43-42 n+9 n^2) a[-3+n]-(-1+n) (-3+2 n) (-1+2 n) (-104+501 n-441 n^2+108 n^3) a[-2+n]-9 (-1+n) (-1+2 n) (-7+3 n) (2-4 n+3 n^2) a[-1+n]+3 (-7+3 n) (-4+3 n) a[n]==0,a[1]==0,a[2]==1,a[3]==70,a[4]==19355},a,{n,1,15}]}] (* Vaclav Kotesovec, Mar 11 2014 *)
terms = 14;
egf = HypergeometricPFQ[{1/6, 5/6}, {}, 12x/(x^2 + 8x + 4)^(3/2)] Exp[-Log[ 1/4 x^2 + 2x + 1]/4 - x/3 + (x^2 + 8x + 4)^(3/2)/(24x) - 1/(3x) - x^2/24 - 1] + O[x]^terms;
CoefficientList[egf, x] (2 Range[0, terms-1])! (* Jean-François Alcover, Nov 23 2018, after Mark van Hoeij *)

a(n) = sum(i=0, 2*n, sum(k=0, min(floor((3*n-i)/3), floor((2*n-i)/2)), sum(j=0, min(floor((3*n-i-3*k)/2), floor((2*n-i-2*k)/2)), ((-1)^(i+j)*(2*n)!*(2*(3*n-i-2*j-3*k))!)/(2^(5*n-i-2*j-4*k)*3^(2*n-i-2*j-k)*(3*n-i-2*j-3*k)!*i!*j!*k!*(2*n-i-2*j-2*k)!)))); \\ Michel Marcus, Jan 18 2018

From Vladeta Jovovic, Mar 25 2001: (Start)
E.g.f. f(x) = Sum_{n >= 0} a(2 * n) * x^n/(2 * n)! satisfies differential equation 6 * x^2 * (-x^2 - 2 * x + 2) * (d^2/dx^2)f(x) - (x^5 + 6 * x^4 + 6 * x^3 - 32 * x + 8) * (d/dx)f(x) + (x/6) * (-x^2 - 2 * x + 2)^2 * f(x) = 0.
Recurrence: a(2 * n) = (2 * n)!/n! * v(n) where 48 * v(n) + (-72 * n^2 + 24 * n + 48) * v(n - 1) + (72 * n^3 - 432 * n^2 + 788 * n - 428) * v(n - 2) + (36 * n^4 - 324 * n^3 + 1052 * n^2 - 1428 * n + 664) * v(n - 3) + (36 * n^4 - 360 * n^3 + 1260 * n^2 - 1800 * n + 864) * v(n - 4) + (6 * n^5 - 94 * n^4 + 550 * n^3 - 1490 * n^2 + 1844 * n - 816) * v(n - 5) + (-n^5 + 15 * n^4 - 85 * n^3 + 225 * n^2 - 274 * n + 120) * v(n - 6) = 0. (End)
a(n) = Sum_{i=0..2*n} Sum_{k=0..min(floor((3*n-i)/3), floor((2*n-i)/2))} Sum_{j=0..min(floor((3*n-i-3*k)/2), floor((2*n-i-2*k)/2))} ((-1)^(i+j)*(2*n)!*(2*(3*n-i-2*j-3*k))!)/(2^(5*n-i-2*j-4*k)*3^(2*n-i-2*j-k)*(3*n-i-2*j-3*k)!*i!*j!*k!*(2*n-i-2*j-2*k)!). - Shanzhen Gao, Jun 05 2009
E.g.f.: hypergeom([1/6, 5/6],[],12*x/(x^2+8*x+4)^(3/2))*exp(-log(1/4*x^2+2*x+1)/4 - x/3 + (x^2+8*x+4)^(3/2)/(24*x) - 1/(3*x) - x^2/24 - 1). Multiply x^i by (2*i)! to get the generating function. - Mark van Hoeij, Nov 07 2011
From Vaclav Kotesovec, Mar 11 2014: (Start)
D-finite with recurrence: 3*(3*n-7)*(3*n-4)*a(n) = 9*(n-1)*(2*n-1)*(3*n-7)*(3*n^2 - 4*n + 2)*a(n-1) + (n-1)*(2*n-3)*(2*n-1)*(108*n^3 - 441*n^2 + 501*n - 104)*a(n-2) + 2*(n-2)*(n-1)*(2*n-5)*(2*n-3)*(2*n-1)*(3*n-1)*(9*n^2 - 42*n + 43)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*(2*n-7)*(2*n-5)*(2*n-3)*(2*n-1)*(3*n-4)*(3*n-1)*a(n-4).
a(n) ~ sqrt(2) * 6^n * n^(3*n) / exp(3*n+2). (End)

More terms from Vladeta Jovovic, Mar 25 2001

A324163 Triangle read by rows: T(n,k) is the number of connected k-regular simple graphs on n labeled vertices, (0 <= k < n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 12, 0, 1, 0, 0, 60, 70, 15, 1, 0, 0, 360, 0, 465, 0, 1, 0, 0, 2520, 19320, 19355, 3507, 105, 1, 0, 0, 20160, 0, 1024380, 0, 30016, 0, 1, 0, 0, 181440, 11166120, 66462480, 66462606, 11180820, 286884, 945, 1

Offset: 1

Andrew Howroyd, Sep 02 2019

			Triangle begins:
  1;
  0, 1;
  0, 0,     1;
  0, 0,     3,     1;
  0, 0,    12,     0,       1;
  0, 0,    60,    70,      15,    1;
  0, 0,   360,     0,     465,    0,     1;
  0, 0,  2520, 19320,   19355, 3507,   105, 1;
  0, 0, 20160,     0, 1024380,    0, 30016, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210

Column k=2 is A001710(n-1) for n >= 3.
Column k=3 is aerated A004109.
Column k=4 is A272905.
Row sums are A322659.
Cf. A059441 (not necessarily connected), A068934 (unlabeled).

Column k is the logarithmic transform of column k of A059441.

A006714 Number of trivalent bipartite labeled graphs with 2n labeled nodes.

Original entry on oeis.org

10, 840, 257040, 137260200, 118273755600, 154712104747200, 292311804557572800, 766931112143320924800, 2706462791802644002128000, 12512595130808078973370704000, 74130965352250071944327288640000, 552334353713465817349513210512960000, 5092566798555894395129552704613028960000

Offset: 3

N. J. A. Sloane

nonn

R. C. Read incorrectly has a(7) = 118257539400 and a(8) = 154678050727200 which he calculated by hand. - Sean A. Irvine, Jun 27 2017

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 3..50
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)

Cf. A001501, A002829, A004109, A246599.

(* b stands for A001501 *) b[n_] := n!^2 Sum[2^(2k-n) 3^(k-n) (3(n-k))!  HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k)!^2), {k, 0, n}]/6^n;
(* c stands for A246599 *) c[n_] := c[n] = Binomial[2n-1, n] b[n] - Sum[ Binomial[2n-1, 2k] Binomial[2k, k] b[k] c[n-k], {k, 1, n-1}];
a[n_] := a[n] = c[n] + Sum[Binomial[2n-1, 2k-1] c[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

\\ here b(n) is A001501
b(n) = {n!^2 * sum(j=0, n, sum(i=0, n-j, my(k=n-i-j); (j + 3*k)! / (3^i * 36^k * i! * k!^2)) / (j! * (-2)^j))}
seq(n)={my(v=vector(n,n,b(n)*binomial(2*n-1,n)), u=vector(n), s=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, 2*binomial(2*n-1,2*k)*v[k]*u[n-k]); s[n]=u[n] + sum(k=3, n-3, binomial(2*n-1,2*k-1)*u[k]*s[n-k])); s[3..n]} \\ Andrew Howroyd, May 22 2018

a(n) = A246599(n) + Sum_{k=1..n-1} binomial(2*n-1,2*k-1)*A246599(k)*a(n-k). - Andrew Howroyd, May 22 2018

a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n+2)). - Vaclav Kotesovec, Feb 17 2024

a(7)-a(8) corrected and a(9)-a(12) computed with nauty by Sean A. Irvine, Jun 27 2017

Terms a(13) and beyond from Andrew Howroyd, May 22 2018

A007099 Number of labeled trivalent (or cubic) 2-connected graphs with 2n nodes.

Original entry on oeis.org

0, 1, 70, 19320, 11052720, 11408720400, 19285018552800, 49792044478176000, 186348919238786304000, 970566620767088881536000, 6808941648018137282054400000, 62642603299257346706851910400000

Offset: 1

N. J. A. Sloane

nonn

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
R. W. Robinson, Computer print-out, no date. Gives first 29 terms.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Table of n, a(n) for n = 1..29 (corrected by Michel Marcus, Jan 19 2019)
G.-B. Chae, E. M. Palmer, R. W. Robinson, Counting labeled general cubic graphs, Discr. Math. 307 (2007) 2979-2992, eqs. (23) and (24).
R. W. Robinson, Cubic labeled graphs, computer print-out, n.d.

Cf. A002829, A004109.

s := proc(n)
    option remember;
    if n = 1 then
        0;
    elif n = 2 then
        1;
    else
        3*n*procname(n-1)+2*procname(n-2)+(3*n-1)*add(procname(i)*procname(n-1-i),i=2..n-3) ;
    end if;
end proc:
A007099 := proc(n)
    if n = 1 then
        0;
    elif n = 2 then
        1;
    else
        (2*n)!/3/n/2^n*(s(n)-2*s(n-1)) ;
    end if;
end proc: # R. J. Mathar, Nov 08 2018

s[n_] := s[n] = If[n <= 2, n - 1, 3 n s[n - 1] + 2 s[n - 2] + (3 n - 1) Sum[s[i] s[n - 1 - i], {i, 2, n - 3}]]; Array[Floor[(2 #)!*(s[#] - 2 s[# - 1])/(3 # 2^#)] &, 12] (* Michael De Vlieger, Oct 11 2017 *)

a(n) = (2*n)! * (s(n) - 2*s(n-1)) / (3*n*2^n) where s(1)=0, s(2)=1, and s(n) = 3*n*s(n-1) + 2*s(n-2) + (3*n-1) * Sum_{i=2..n-3} s(i) * s(n-1-i). - Sean A. Irvine, Oct 11 2017

A246599 Number of connected trivalent bipartite labeled graphs with 2n labeled nodes.

Original entry on oeis.org

10, 840, 257040, 137214000, 118248530400, 154686980448000, 292276881344448000, 766864651478365440000, 2706292794907249067520000, 12512021073989410699165440000, 74128448237031250090060032000000, 552320243355746711191770103680000000, 5092467146398443040845772685937408000000

Offset: 3

N. J. A. Sloane, Sep 08 2014

nonn

R. C. Read incorrectly has a(7) = 118237555800 and a(8) = 154652926428000 which he calculated by hand.

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

Andrew Howroyd, Table of n, a(n) for n = 3..50
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)

Cf. A001501, A002829, A004109, A006714.

b[n_] := n!^2*Sum[2^(2k-n) 3^(k-n)(3(n-k))!*HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k )!^2), {k, 0, n}]/6^n;
a[n_] := a[n] = Binomial[2n-1, n] b[n] - Sum[Binomial[2n-1, 2k] Binomial[2 k, k] b[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

\\ here b(n) is A001501
b(n) = {n!^2 * sum(j=0, n, sum(i=0, n-j, my(k=n-i-j); (j + 3*k)! / (3^i * 36^k * i! * k!^2)) / (j! * (-2)^j))}
seq(n)={my(v=vector(n, n, b(n)*binomial(2*n, n)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, binomial(2*n-1,2*k)*v[k]*u[n-k])); u[3..n]/2} \\ Andrew Howroyd, May 22 2018

a(n) = binomial(2*n-1, n)*A001501(n) - Sum_{k=1..n-1} binomial(2*n-1, 2*k) * binomial(2*k, k) * A001501(k) * a(n-k). - Andrew Howroyd, May 22 2018

a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n + 2)). - Vaclav Kotesovec, Feb 17 2024

a(7)-a(8) corrected and a(9)-a(12) computed with nauty by Sean A. Irvine, Jun 27 2017

Terms a(13) and beyond from Andrew Howroyd, May 22 2018

cp's OEIS Frontend

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

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A002829 Number of trivalent (or cubic) labeled graphs with 2n nodes.

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A324163 Triangle read by rows: T(n,k) is the number of connected k-regular simple graphs on n labeled vertices, (0 <= k < n).

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A006714 Number of trivalent bipartite labeled graphs with 2n labeled nodes.

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A007099 Number of labeled trivalent (or cubic) 2-connected graphs with 2n nodes.

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Maple

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A246599 Number of connected trivalent bipartite labeled graphs with 2n labeled nodes.

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Mathematica

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A007102 Number of labeled disconnected trivalent (or cubic) graphs with 2n nodes.

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