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Search: a185135 -id:a185135
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A005638
Number of unlabeled trivalent (or cubic) graphs with 2n nodes.
(Formerly M1656)
+10
31
1, 0, 1, 2, 6, 21, 94, 540, 4207, 42110, 516344, 7373924, 118573592, 2103205738, 40634185402, 847871397424, 18987149095005, 454032821688754, 11544329612485981, 310964453836198311, 8845303172513781271
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Because the triangle A051031 is symmetric, a(n) is also the number of (2n-4)-regular graphs on 2n vertices.
REFERENCES
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Table of n, a(n) for n=0..20.
G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Jason Kimberley, Not-necessarily connected regular graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
R. W. Robinson, Cubic graphs (notes)
Robinson, R. W.; Wormald, N. C., Numbers of cubic graphs, J. Graph Theory 7 (1983), no. 4, 463-467.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Cubic Graph
Gal Weitz, Lirandë Pira, Chris Ferrie, and Joshua Combes, Sub-universal variational circuits for combinatorial optimization problems, arXiv:2308.14981 [quant-ph], 2023.
FORMULA
a(n) = A002851(n) + A165653(n).
This sequence is the Euler transformation of A002851.
CROSSREFS
Cf. A000421.
Row sums of A275744.
3-regular simple graphs: A002851 (connected), A165653 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), this sequence (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Not necessarily connected 3-regular simple graphs with girth *at least* g: this sequence (g=3), A185334 (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Ronald C. Read.
Comment, formulas, and (most) crossrefs by Jason Kimberley, 2009 and 2012
STATUS
approved
A185133
Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 3.
+10
14
0, 0, 1, 1, 4, 15, 71, 428, 3406, 34270, 418621, 5937051, 94782437, 1670327647, 32090011476, 666351752261, 14859579573845
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..16.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
FORMULA
a(n) = A005638(n) - A185334(n).
a(n) = A006923(n) + A185033(n).
CROSSREFS
Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), this sequence (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).
Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: this sequence (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
KEYWORD
nonn,hard,more
AUTHOR
Jason Kimberley, Mar 21 2012
STATUS
approved
A185334
Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 4.
+10
13
1, 0, 0, 1, 2, 6, 23, 112, 801, 7840, 97723, 1436873, 23791155, 432878091, 8544173926, 181519645163, 4127569521160
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The null graph on 0 vertices is vacuously 3-regular; since it is acyclic, it has infinite girth.
LINKS
Table of n, a(n) for n=0..16.
Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
FORMULA
Euler transformation of A014371.
MATHEMATICA
A014371 = Cases[Import["https://oeis.org/A014371/b014371.txt", "Table"], {_, _}][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014371] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)
CROSSREFS
3-regular simple graphs with girth at least 4: A014371 (connected), A185234 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: A008484 (k=2), this sequence (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), this sequence (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
KEYWORD
nonn,more,hard
AUTHOR
Jason Kimberley, Feb 15 2011
STATUS
approved
A185134
Number of, not necessarily connected, 3-regular simple graphs on 2n vertices with girth exactly 4.
+10
12
0, 0, 0, 1, 2, 5, 21, 103, 752, 7385, 91939, 1345933, 22170664, 401399440, 7887389438, 166897766824, 3781593764772
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..16.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
FORMULA
a(n) = A185334(n) - A185335(n).
a(n) = A006924(n) + A185034(n).
CROSSREFS
Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), A185644 (triangle); fixed k: A026797 (k=2), this sequence (k=3), A185144 (k=4).
Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: A185133 (g=3), this sequence (g=4), A185135 (g=5), A185136 (g=6).
KEYWORD
nonn,hard,more
AUTHOR
Jason Kimberley, Mar 21 2012
STATUS
approved
A185335
Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 5.
+10
9
1, 0, 0, 0, 0, 1, 2, 9, 49, 455, 5784, 90940, 1620491, 31478651, 656784488, 14621878339, 345975756388
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Table of n, a(n) for n=0..16.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
FORMULA
This sequence is the Euler transformation of A014372.
MATHEMATICA
A014372 = Cases[Import["https://oeis.org/A014372/b014372.txt", "Table"], {_, _}][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014372] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)
CROSSREFS
3-regular simple graphs with girth at least 5: A014372 (connected), A185235 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), A185334 (g=4), this sequence (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: A185325 (k=2), this sequence (k=3).
KEYWORD
nonn,more,hard
AUTHOR
Jason Kimberley, Jan 28, 2011
STATUS
approved
A185136
Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 6.
+10
8
0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 32, 385, 7573, 181224, 4624481, 122089999, 3328899592, 93988909792
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
LINKS
Table of n, a(n) for n=0..17.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
FORMULA
a(n) = A006926(n) + A185036(n).
CROSSREFS
Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: A185133 (g=3), A185134 (g=4), A185135 (g=5), this sequence (g=6).
KEYWORD
nonn,hard,more
AUTHOR
Jason Kimberley, Mar 21 2012
STATUS
approved
A185130
Irregular triangle E(n,g) counting not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g.
+10
4
1, 1, 1, 4, 2, 15, 5, 1, 71, 21, 2, 428, 103, 8, 1, 3406, 752, 48, 1, 34270, 7385, 450, 5, 418621, 91939, 5752, 32, 5937051, 1345933, 90555, 385, 94782437, 22170664, 1612917, 7573, 1, 1670327647, 401399440, 31297424, 181224, 3, 32090011476, 7887389438
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,4
COMMENTS
The first column is for girth exactly 3. The column for girth exactly g begins when 2n reaches A000066(g).
LINKS
Table of n, a(n) for n=2..44.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
FORMULA
The n-th row is the sequence of differences of the n-th row of A185330:
E(n,g) = A185330(n,g) - A185330(n,g+1), once we have appended 0 to each row of A185330.
Hence the sum of the n-th row is A185330(n,3) = A005638(n).
EXAMPLE
1;
1, 1;
4, 2;
15, 5, 1;
71, 21, 2;
428, 103, 8, 1;
3406, 752, 48, 1;
34270, 7385, 450, 5;
418621, 91939, 5752, 32;
5937051, 1345933, 90555, 385;
94782437, 22170664, 1612917, 7573, 1;
1670327647, 401399440, 31297424, 181224, 3;
32090011476, 7887389438, 652159986, 4624481, 21;
666351752261, 166897766824, 14499787794, 122089999, 545, 1;
14859579573845, 3781593764772, 342646826428, 3328899592, 30368, 0;
CROSSREFS
Initial columns of this triangle: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
KEYWORD
nonn,hard,tabf
AUTHOR
Jason Kimberley, Dec 26 2012
STATUS
approved
A185336
Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 6.
+10
4
1, 0, 0, 0, 0, 0, 0, 1, 1, 5, 32, 385, 7574, 181227, 4624502, 122090545, 3328929960, 93990692632, 2754222605808
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
COMMENTS
The null graph on 0 vertices is vacuously 3-regular; since it is acyclic, it has infinite girth.
LINKS
Table of n, a(n) for n=0..18.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
FORMULA
Euler transformation of A014374.
MATHEMATICA
A014374 = Cases[Import["https://oeis.org/A014374/b014374.txt", "Table"], {_, _}][[All, 2]];
etr[f_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d f[d], {d, Divisors[j]}] b[n - j], {j, 1, n}]/n]; b];
a = etr[A014374[[# + 1]]&];
a /@ Range[0, Length[A014374] - 1] (* Jean-François Alcover, Dec 04 2019 *)
CROSSREFS
3-regular simple graphs with girth at least 6: A014374 (connected), A185236 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 6: A185326 (k=2), this sequence (k=3).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), A185334 (g=4), A185335 (g=5), this sequence (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
KEYWORD
nonn,more,hard
AUTHOR
Jason Kimberley, Jan 28 2012
EXTENSIONS
a(18) from A014374 from Jean-François Alcover, Dec 04 2019
STATUS
approved
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