A295193
Number of regular simple graphs on n labeled nodes.
Original entry on oeis.org
1, 2, 2, 8, 14, 172, 932, 45936, 1084414, 155862512, 10382960972, 6939278572096, 2203360500122300, 4186526756621772344, 3747344008241368443820, 35041787059691023579970848, 156277111373303386104606663422, 4142122641757598618318165240180096
Offset: 1
From _Gus Wiseman_, Dec 19 2018: (Start)
A graph is regular if all vertices have the same degree. For example, the a(4) = 8 simple regular graphs are:
1 2
3 4
.
4---1 3---1 2---1
3---2 4---2 4---3
.
3---4 4---3 4---2
| | | | | |
1---2 1---2 1---3
.
4---3
| X |
2---1
(End)
- Andrew Howroyd, Table of n, a(n) for n = 1..24
- E. A. Bender and E. R. Canfield, The asymptotic number of labeled graphs with given degree sequences, Journal of Combinatorial Theory, Series A, 24 (1978), 296-307.
- Andrew Howroyd, PARI Program
- Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
- B. D. McKay, Applications of a technique for labelled enumeration, Congress. Numerantium, 40 (1983), 207-221.
- Wikipedia, Regular graph
-
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 *)
-
\\ See link for program file.
for(n=1, 10, print1(A295193(n), ", ")) \\ Andrew Howroyd, Aug 28 2019
A059441
Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 12, 0, 1, 1, 15, 70, 70, 15, 1, 1, 0, 465, 0, 465, 0, 1, 1, 105, 3507, 19355, 19355, 3507, 105, 1, 1, 0, 30016, 0, 1024380, 0, 30016, 0, 1, 1, 945, 286884, 11180820, 66462606, 66462606, 11180820, 286884, 945, 1
Offset: 1
1;
1, 1;
1, 0, 1;
1, 3, 3, 1;
1, 0, 12, 0, 1;
1, 15, 70, 70, 15, 1;
1, 0, 465, 0, 465, 0, 1;
1, 105, 3507, 19355, 19355, 3507, 105, 1;
1, 0, 30016, 0, 1024380, ...;
1, 945, 286884, 11180820, 66462606, ...;
1, 0, 3026655, 0, 5188453830, ...;
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
- Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)
- Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
- Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221. See page 216.
- Wikipedia, Regular graph
-
Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{n,9},{k,0,n-1}] (* Gus Wiseman, Dec 24 2018 *)
-
for(n=1, 10, print(A059441(n))) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019
A322698
Number of regular graphs with half-edges on n labeled vertices.
Original entry on oeis.org
1, 2, 4, 10, 40, 278, 3554, 84590, 3776280, 317806466, 50710452574, 15414839551538, 8964708979273634, 10008446308186072290, 21518891146915893435358, 89320970210116481106835986, 717558285660687970023516336792, 11176382741327158622885664697124082, 338202509574712032788035618665293979610
Offset: 0
The a(3) = 10 edge sets:
{}
{{1},{2,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2},{3}}
{{1,2},{1,3},{2,3}}
{{1},{3},{1,2},{2,3}}
{{1},{2},{1,3},{2,3}}
{{2},{3},{1,2},{1,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}
-
Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Union/@Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,6}]
-
for(n=1, 10, print1(A322698(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019
A322659
Number of connected regular simple graphs on n labeled vertices.
Original entry on oeis.org
1, 1, 1, 4, 13, 146, 826, 44808, 1074557, 155741296, 10381741786, 6939251270348, 2203360264480750, 4186526735251514044, 3747344007864300197810, 35041787059621536192399824, 156277111373298355107598128061, 4142122641757597729416733678931968
Offset: 1
-
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[If[n==1,1,Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],SameQ@@Length/@Split[Sort[Join@@#]],Length[csm[#]]==1]&]]],{n,6}]
A322704
Number of regular hypergraphs on n labeled vertices with no singletons.
Original entry on oeis.org
1, 1, 2, 4, 80, 209944
Offset: 0
The a(3) = 4 edge-sets:
{}
{{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
-
Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2,n}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2^n-n-1}],{n,1,5}]
Showing 1-5 of 5 results.
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