A125671 -id:A125671 - cp's OEIS frontend

A112410 Number of connected simple graphs with n vertices, n+1 edges, and vertex degrees no more than 4.

0, 0, 0, 1, 5, 17, 56, 182, 573, 1792, 5533, 16977, 51652, 156291, 470069, 1407264, 4193977, 12451760, 36838994, 108656009, 319583578, 937634011, 2744720126, 8018165821, 23379886511, 68056985580, 197800670948, 574068309840, 1663907364480, 4816910618093, 13929036720057

Offset: 1

Jonathan Vos Post, Dec 08 2005

nonn

Such graphs are also referred to (e.g., by Hendrickson & Parks) as carbon skeletons with two rings, or bicyclic skeletons, although actual number of simple cycles in such graphs can exceed 2 (e.g., in the example). - Andrey Zabolotskiy, Nov 24 2017

Terms computed with nauty agree at least to a(20) with those computed by formula and sequences A125669, A125670, A125671, A305132. - Andrew Howroyd, May 26 2018

			The only such graph for n = 4 is:
o-o
|/|
o-o

Andrew Howroyd, Table of n, a(n) for n = 1..200
J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 2 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs.

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, this sequence, A112619, A112408, A112424, A112425, A112426, A112442.
Cf. A121941 (any number of edges), A006820 (2n edges).
Cf. A125669, A125670, A125671, A305132.

for n in {4..15}; do geng -c -D4 ${n} $((n+1)):$((n+1)) -u; done # Andrey Zabolotskiy, Nov 24 2017

a(n) = A125669(n) + A125670(n) + A125671(n) + A305132(n). - Andrew Howroyd, May 26 2018

Corrected offset and new name from Andrey Zabolotskiy, Nov 20 2017
a(20) corrected by Andrey Zabolotskiy and Andrew Howroyd, May 26 2018
Terms a(21) and beyond from Andrew Howroyd, May 26 2018

A121165 Number of fused bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

Original entry on oeis.org

1, 2, 7, 15, 44, 107, 295, 763, 2077, 5533, 15053, 40697, 111028, 302583, 828176, 2267939, 6225340, 17103834, 47062513, 129616014, 357364708, 986110340, 2723373330, 7526669582, 20816208417, 57606623093, 159514679011, 441942381946, 1225049208597, 3397418545998

Offset: 4

Parthasarathy Nambi, Aug 13 2006

nonn

Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles of the same length joined at a single edge and all nodes having degree at most 4. - Andrew Howroyd, May 25 2018

Andrew Howroyd, Table of n, a(n) for n = 4..200
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).

Cf. A125671.

\\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
C1(n)={sum(k=1, n\2, d1^(2*k) + d2^k)/4}
C2(n)={sum(k=1, n\2, d2^k + d2^(k-k%2)*d1^(2*(k%2)))/4}
seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2],[g(d,1),g(d,2)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d4], [g(d,1),g(d,2),g(d,4)]))} \\ Andrew Howroyd, May 25 2018

More terms from N. J. A. Sloane, Aug 27 2006

Terms a(26) and beyond from Andrew Howroyd, May 25 2018

A305132 Number of connected graphs on n unlabeled nodes with exactly 2 cycles joined along two or more edges but not more than half each cycle and all nodes having degree at most 4.

Original entry on oeis.org

1, 3, 11, 36, 116, 366, 1151, 3583, 11093, 34141, 104489, 318139, 963899, 2907276, 8731919, 26125538, 77889504, 231466147, 685811867, 2026481941, 5973064855, 17565416721, 51547293439, 150977445294, 441409701444, 1288409915625, 3754926609800, 10927779696264

Offset: 5

Andrew Howroyd, May 26 2018

nonn

The resulting graph will actually have three cycles. See A121331 for the special case of all three cycles having the same length.

Equivalently, the number of connected simple graphs with n unlabeled nodes and n + 1 edges and all nodes having degree at most 4 (A112410) less those graphs described by A125669, A125670 and A125671.

			Illustration of graphs for n=5 and n=6:
    o          o--o       o           o--o
   /|\        /|\        /|\         /|  |
  o o o      o o o      o o o--o    o o  |
   \|/        \|/        \|/         \|  |
    o          o          o           o--o

Andrew Howroyd, Table of n, a(n) for n = 5..200

Cf. A112410, A121331, A125669, A125670, A125671, A125672, A125673.

\\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
C1(n)={subst(Pol(x^3*d1^3/(1-x*d1)^3 + 3*x^3*d1*d2/((1-x*d1)*(1-x^2*d2)) + 2*x^3*d3/(1-x^3*d3) + O(x*x^n)), x, 1)/12}
C2(n)={subst(Pol(((x*d1+x^2*d2)/(1-x^2*d2))^3 + 3*(x*d1+x^2*d2)*x^2*d2/(1-x^2*d2)^2 + 2*(x^3*d3 + x^6*d6)/(1-x^6*d6) + O(x*x^n)), x, 1)/12}
seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2,d3],[g(d,1), g(d,2), g(d,3)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d3,d6], [g(d,1), g(d,2), g(d,3), g(d,6)]))}

a(n) >= A125672(n) + A125673(n).

cp's OEIS Frontend

A112410 Number of connected simple graphs with n vertices, n+1 edges, and vertex degrees no more than 4.

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A121165 Number of fused bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

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A305132 Number of connected graphs on n unlabeled nodes with exactly 2 cycles joined along two or more edges but not more than half each cycle and all nodes having degree at most 4.

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