A295193 - OEIS

%I #36 Apr 22 2021 17:56:47

%S 1,2,2,8,14,172,932,45936,1084414,155862512,10382960972,6939278572096,

%T 2203360500122300,4186526756621772344,3747344008241368443820,

%U 35041787059691023579970848,156277111373303386104606663422,4142122641757598618318165240180096

%N Number of regular simple graphs on n labeled nodes.

%H Andrew Howroyd, <a href="/A295193/b295193.txt">Table of n, a(n) for n = 1..24</a>

%H E. A. Bender and E. R. Canfield, <a href="https://doi.org/10.1016/0097-3165(78)90059-6">The asymptotic number of labeled graphs with given degree sequences</a>, Journal of Combinatorial Theory, Series A, 24 (1978), 296-307.

%H Andrew Howroyd, <a href="/A295193/a295193.txt">PARI Program</a>

%H Atabey Kaygun, <a href="https://arxiv.org/abs/2101.02299">Enumerating Labeled Graphs that Realize a Fixed Degree Sequence</a>, arXiv:2101.02299 [math.CO], 2021.

%H B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/papers/LabelledEnumeration.pdf">Applications of a technique for labelled enumeration</a>, Congress. Numerantium, 40 (1983), 207-221.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Regular_graph">Regular graph</a>

%e From _Gus Wiseman_, Dec 19 2018: (Start)

%e A graph is regular if all vertices have the same degree. For example, the a(4) = 8 simple regular graphs are:

%e   1 2

%e   3 4

%e .

%e   4---1  3---1  2---1

%e   3---2  4---2  4---3

%e .

%e   3---4  4---3  4---2

%e   |   |  |   |  |   |

%e   1---2  1---2  1---3

%e .

%e   4---3

%e   | X |

%e   2---1

%e (End)

%t Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,9}] (* _Gus Wiseman_, Dec 19 2018 *)

%o (PARI) \\ See link for program file.

%o for(n=1, 10, print1(A295193(n), ", ")) \\ _Andrew Howroyd_, Aug 28 2019

%Y Row sums of A059441.

%Y Cf. A005176 (unlabeled equivalent), A058891, A116539, A299353, A306017, A306021, A319189, A319190, A319612, A319729.

%K nonn

%O 1,2

%A _Álvar Ibeas_, Nov 16 2017

%E a(16)-a(18) from _Andrew Howroyd_, Aug 28 2019