A112410 -id:A112410 - cp's OEIS frontend

A112408 Number of connected simple graphs with n vertices, n+3 edges, and vertex degrees no more than 4.

0, 0, 0, 0, 2, 14, 79, 430, 2161, 10162, 45282, 192945, 790849, 3138808, 12116550, 45675153, 168661704, 611701138, 2183635232, 7686541342, 26720976964, 91856241351, 312594121721, 1054104924270

Offset: 1

Jonathan Vos Post, Dec 21 2005

nonn

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 4 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, A112410, A112619, this sequence, A112424, A112425, A112426, A112442. Cf. A121941.

for n in {5..15}; do geng -c -D4 ${n} $((n+3)):$((n+3)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset, new name, and a(18) from Andrey Zabolotskiy, Nov 24 2017

a(18)-a(24) added by Georg Grasegger, Jun 05 2023

A112424 Number of connected simple graphs with n vertices, n+4 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 59, 427, 2768, 16461, 90111, 460699, 2222549, 10216607, 45076266, 192059940, 794088479, 3198709835, 12593964702, 48596474890, 184195614359, 687087962550, 2526421534903

Offset: 1

Jonathan Vos Post, Dec 21 2005

nonn

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 5 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, A112410, A112619, A112408, this sequence, A112425, A112426, A112442. Cf. A121941.

for n in {5..15}; do geng -c -D4 ${n} $((n+4)):$((n+4)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset and new name from Andrey Zabolotskiy, Nov 24 2017

a(16)-a(23) added by Georg Grasegger, Jun 05 2023

A112425 Number of connected simple graphs with n vertices, n+5 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 31, 298, 2616, 20346, 140605, 880737, 5082279, 27402524, 139587885, 677772953, 3158930531, 14212444473, 62009204208, 263350765116, 1092085621098, 4433596269478

Offset: 1

Jonathan Vos Post, Dec 21 2005

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 6 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. [gives different a(15)]

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, A112619, A112408, A112424, this sequence, A112426, A112442. Cf. A121941.

for n in {5..13}; do geng -c -D4 ${n} $((n+5)):$((n+5)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset and new name from Andrey Zabolotskiy, Nov 24 2017

a(15) corrected and a(16)-a(22) added by Georg Grasegger, Jun 05 2023

A112426 Number of connected simple graphs with n vertices, n+6 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 9, 134, 1714, 18436, 167703, 1327240, 9372119, 60324933, 359730035, 2012733260, 10670975762, 54028108819, 262872075003, 1235323112178, 5630370812614

Offset: 1

Jonathan Vos Post, Dec 21 2005

Distribution of carbon skeletons. See the paper by Hendrikson and Parks for details. If n=6 the number of 7-cyclic skeletons is 1. If n=7 the number of 7-cyclic skeletons is 9. If n=10 the number of 7-cyclic skeletons is 18436. - Parthasarathy Nambi, Jan 05 2007

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 7 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs [gives different numbers for n > 10].

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, A112619, A112408, A112424, A112425, this sequence, A112442. Cf. A121941.

for n in {6..13}; do geng -c -D4 ${n} $((n+6)):$((n+6)) -u; done # Andrey Zabolotskiy, Nov 24 2017

New name, offset corrected, and a(11)-a(14) corrected by Andrey Zabolotskiy, Nov 24 2017

a(15)-a(21) added by Georg Grasegger, Jun 05 2023

A112619 Number of connected simple graphs with n vertices, n+2 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 1, 4, 18, 79, 326, 1278, 4875, 17978, 64720, 227842, 787546, 2678207, 8982754, 29761361, 97558039, 316778169, 1019996738, 3259673935, 10347077497, 32644696187, 102425388286, 319754805262

Offset: 1

Jonathan Vos Post, Dec 21 2005

nonn

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 3 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. [gives different a(18)]

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, this sequence, A112408, A112424, A112425, A112426, A112442. Cf. A121941.

for n in {4..15}; do geng -c -D4 ${n} $((n+2)):$((n+2)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset and new name from Andrey Zabolotskiy, Nov 24 2017

a(18) corrected and a(19)-a(25) added by Georg Grasegger, Jun 05 2023

A112442 Number of connected simple graphs with n vertices, n+7 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 35, 707, 11477, 146428, 1530906, 13663758, 107554370, 764873164, 5004170844, 30537798974, 175688807383, 960958921848, 5030916734826

Offset: 1

Jonathan Vos Post, Dec 21 2005

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 8 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs [gives different a(11)].

The analogs for n+k edges with k = -1, 0, ..., 6 are: A000602, A036671, A112410, A112619, A112408, A112424, A112425, A112426. Cf. A121941.

for n in {7..13}; do geng -c -D4 ${n} $((n+7)):$((n+7)) -u; done # Andrey Zabolotskiy, Nov 24 2017

New name, offset corrected, a(11) corrected, and a(14) added by Andrey Zabolotskiy, Nov 24 2017

a(15)-a(20) added by Georg Grasegger, Jun 05 2023

A125064 Number of simple graphs on at most 16 unlabeled vertices with maximal degree at most 4 with a single cycle of length 16-n.

Original entry on oeis.org

1, 2, 11, 39, 169, 534, 1612, 3894, 8771, 16307, 29391, 43291, 69429, 83571

Offset: 0

Parthasarathy Nambi, Jan 05 2007

In the terms of the paper by Hendrickson and Parks, a(n) is the number of monocyclic skeletons with up to 16 nodes with a ring of size 16-n.

James B. Hendrickson and Camden A. Parks, Generation and Enumeration of Carbon Skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), pp. 101-107. See Table VI on page 103.

Cf. A000602, A036671, A112410, A112619, A112408, A112424, A112425, A112426.

Sum_n a(n) = Sum_{k=3..16} A036671(k).

Edited by Andrey Zabolotskiy, Feb 02 2025

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).

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A112408 Number of connected simple graphs with n vertices, n+3 edges, and vertex degrees no more than 4.

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A112424 Number of connected simple graphs with n vertices, n+4 edges, and vertex degrees no more than 4.

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A112425 Number of connected simple graphs with n vertices, n+5 edges, and vertex degrees no more than 4.

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A112426 Number of connected simple graphs with n vertices, n+6 edges, and vertex degrees no more than 4.

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A112619 Number of connected simple graphs with n vertices, n+2 edges, and vertex degrees no more than 4.

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A112442 Number of connected simple graphs with n vertices, n+7 edges, and vertex degrees no more than 4.

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A125064 Number of simple graphs on at most 16 unlabeled vertices with maximal degree at most 4 with a single cycle of length 16-n.

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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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