A321427
Number of connected labeled closely cubic graphs on 2n+1 nodes.
Original entry on oeis.org
0, 0, 30, 5670, 2543940, 2147121900, 3060711804150, 6822508357214550, 22450423357516354200, 104310014134397398727400, 660475190873012530467201750, 5537072793132139084007288856750, 60005787711473418534665255077267500, 823803200874542135657355819087997282500
Offset: 0
-
nmax = 13;
b[n_] := Sum[Sum[Sum[((-1)^(i + j) (2n)! (2(3n - i - 2j - 3k))!)/(2^(5n - i - 2j - 4k) 3^(2n - i - 2j - k)(3n - i - 2j - 3k)! i! j! k! (2n - i - 2j - 2k)!), {j, 0, Min[Floor[(3n - i - 3k)/2], Floor[(2n - i - 2k)/2]]}], {k, 0, Min[Floor[(3n - i)/3], Floor[(2n - i)/2]]}], {i, 0, 2n}];
seq[n_] := seq[n] = Module[{v = Table[0, {n + 1}]}, For[k = 2, k <= n, k++, v[[k + 1]] = 3k b[k] + 2k(2k - 1) v[[k]] + k(2k - 1)(2k - 2)(2k - 3)v[[k - 1]]]; v];
a[n_] := (2n+1) seq[nmax][[n+1]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 23 2018, after Andrew Howroyd *)
A338978
Number of labeled 5-regular graphs on 2n nodes.
Original entry on oeis.org
1, 0, 0, 1, 3507, 66462606, 2977635137862, 283097260184159421, 52469332407700365320163, 17647883828569858659972268092, 10148613081040117624319536901932188, 9494356410654311931931879706070629989407, 13859154719468565627065764000731047706917194485
Offset: 0
- Marni Mishna, Table of n, a(n) for n = 0..110 (first 51 terms from Brendan Mackay)
- Frédéric Chyzak and Marni Mishna, Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction based approach, arXiv:2406.04753 [math.CO], 2024.
- Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 9.
- Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence.
- Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
With interspersed zeros, column k=5 of
A059441.
A007102
Number of labeled disconnected trivalent (or cubic) graphs with 2n nodes.
Original entry on oeis.org
1, 0, 0, 0, 35, 14700, 11832975, 15245900670, 29683109280825, 84114515340655800, 335974271076054435825, 1839316574841276904122750, 13461678841737111645720135075, 128798406388658994689642297857500
Offset: 0
- R. W. Robinson, personal communication.
- R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A286757
Number of labeled connected rooted trivalent graphs with 2n nodes.
Original entry on oeis.org
0, 4, 120, 33600, 18471600, 18386121600, 30231607606200, 76388992266787200, 281063897503929540000, 1444102677105174358272000, 10020068498645397815029407000, 91355440119583548608158042584000, 1069762020017605579789451640683370000
Offset: 1
- R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
A109542
a(n) = number of labeled 3-regular (trivalent) multi-graphs without self-loops on 2n vertices with a maximum of 2 edges between any pair of nodes. Also a(n) = number of labeled symmetric 2n X 2n matrices with {0,1,2}-entries with row sum equal to 3 for each row and trace 0.
Original entry on oeis.org
0, 7, 640, 170555, 94949400, 95830621425, 159062872168200, 404720953797785625
Offset: 1
a(2)=7 because for 2*n=4 nodes there are 7 possible labeled graphs whose adjacency matrices are as follows:
0 2 1 0
2 0 0 1
1 0 0 2
0 1 2 0;
0 1 2 0
1 0 0 2
2 0 0 1
0 2 1 0;
0 2 0 1
2 0 1 0
0 1 0 2
1 0 2 0;
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0;
0 0 2 1
0 0 1 2
2 1 0 0
1 2 0 0;
0 1 0 2
1 0 2 0
0 2 0 1
2 0 1 0;
0 0 1 2
0 0 2 1
1 2 0 0
2 1 0 0.
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