A051031 -id:A051031 - cp's OEIS frontend

A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 3, 2, 1, 1, 1, 0, 1, 0, 4, 0, 4, 0, 1, 1, 0, 1, 1, 5, 7, 9, 4, 1, 1, 1, 0, 1, 0, 7, 0, 24, 0, 7, 0, 1, 1, 0, 1, 1, 8, 16, 54, 60, 32, 8, 1, 1, 1, 0, 1, 0, 10, 0, 128, 0, 240, 0, 12, 0, 1, 1, 0, 1, 1, 12, 37, 271, 955, 1753, 930, 135, 14, 1, 1

Offset: 0

Andrew Howroyd, Mar 15 2020

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A333351. Burnside's lemma can be used to extend this method to the unlabeled case.

			Array begins:
=================================================
n\k | 0 1 2  3   4    5      6     7        8
----+--------------------------------------------
  0 | 1 1 1  1   1    1      1     1        1 ...
  1 | 1 0 0  0   0    0      0     0        0 ...
  2 | 1 1 1  1   1    1      1     1        1 ...
  3 | 1 0 1  0   1    0      1     0        1 ...
  4 | 1 1 2  3   4    5      7     8       10 ...
  5 | 1 0 2  0   7    0     16     0       37 ...
  6 | 1 1 4  9  24   54    128   271      582 ...
  7 | 1 0 4  0  60    0    955     0    12511 ...
  8 | 1 1 7 32 240 1753  13467 90913   543779 ...
  9 | 1 0 8  0 930    0 253373     0 35255015 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..350

Columns k=0..8 are (with interspersed 0's for odd k): A000012, A000012, A002865, A129416, A129418, A129420, A129422, A129424, A129426.
Row n=4 is A001399.
Cf. A051031 (simple graphs), A167625 (with loops), A192517 (not necessarily regular), A328682 (connected), A333351 (labeled nodes).

A165626 Number of 5-regular graphs (quintic graphs) on 2n vertices.

Original entry on oeis.org

1, 0, 0, 1, 3, 60, 7849, 3459386, 2585136741, 2807105258926, 4221456120848125, 8516994772686533749, 22470883220896245217626, 75883288448434648617038134, 322040154712674550886226182668

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-6)-regular graphs on 2n vertices.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms

5-regular simple graphs: A006821 (connected), A165655 (disconnected), this sequence (not necessarily connected).

Regular graphs A005176 (any degree), A051031 (triangular array), specified degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), this sequence (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).

A006821 = Cases[Import["https://oeis.org/A006821/b006821.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006821] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transform of A006821.

Regular graphs cross-references edited by Jason Kimberley, Nov 07 2009
a(9) from Jason Kimberley, Nov 24 2009
a(10)-a(14) from Andrew Howroyd, Mar 10 2020

A165627 Number of 6-regular graphs (sextic graphs) on n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7849, 367860, 21609301, 1470293676, 113314233813, 9799685588961, 945095823831333, 101114579937196179, 11945375659140003692, 1551593789610531820695, 220716215902794066709555, 34259321384370735003091907, 5782740798229835127025560294

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (n-7)-regular graphs on n vertices.

Georg Grasegger, Hakan Guler, Bill Jackson, Anthony Nixon, Flexible circuits in the d-dimensional rigidity matroid, arXiv:2003.06648 [math.CO], 2020.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Regular Graph
Eric Weisstein's World of Mathematics, Sextic Graph

6-regular simple graphs: A006822 (connected), A165656 (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), A005638 (k=3), A033301 (k=4), A165626 (k=5), this sequence (k=6), A165628 (k=7), A180260 (k=8).

A006822 = Cases[Import["https://oeis.org/A006822/b006822.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006822] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A006822.

Cross-references edited by Jason Kimberley, Nov 07 2009 and Oct 17 2011
a(17) from Jason Kimberley, Dec 30 2010
a(18)-a(24) from Andrew Howroyd, Mar 07 2020

A165628 Number of 7-regular graphs (septic graphs) on 2n vertices.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 1547, 21609301, 733351105935, 42700033549946255, 4073194598236125134140, 613969628444792223023625238, 141515621596238755267618266465449

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-8)-regular graphs on 2n vertices.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Septic Graph

7-regular simple graphs: A014377 (connected), A165877 (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), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), this sequence (k=7), A180260 (k=8).

A014377 = Cases[Import["https://oeis.org/A014377/b014377.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014377] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014377.

Cross-references edited by Jason Kimberley, Nov 07 2009 and Oct 17 2011
a(9)-a(11) from Andrew Howroyd, Mar 09 2020
a(12) from Andrew Howroyd, May 19 2020

A008327 Triangle read by rows: T(n,k) is the number of simple regular bipartite graphs with 2n nodes and degree k, (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 6, 4, 1, 1, 1, 1, 4, 14, 14, 4, 1, 1, 1, 1, 7, 41, 130, 41, 7, 1, 1, 1, 1, 8, 157, 1981, 1981, 157, 8, 1, 1, 1, 1, 12, 725, 62616, 304496, 62616, 725, 12, 1, 1, 1, 1, 14, 4196, 2806508, 78322916

Offset: 0

Brendan McKay

This sequence can be derived from A008326 by Euler transform. - Andrew Howroyd, Apr 03 2020

			Triangle begins:
  1,
  1, 1,
  1, 1, 1,
  1, 1, 1,   1,
  1, 1, 2,   1,    1,
  1, 1, 2,   2,    1,    1,
  1, 1, 4,   6,    4,    1,   1;
  1, 1, 4,  14,   14,    4,   1, 1;
  1, 1, 7,  41,  130,   41,   7, 1, 1;
  1, 1, 8, 157, 1981, 1981, 157, 8, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..189
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Column k=0..5 are A000012, A000012, A002865, A008325, A333730, A333731.
Row sums are A008324.
Cf. A051031, A008326, A087114, A133687, A333159.

Column k is the Euler transform of column k of A008326. - Andrew Howroyd, Apr 03 2020

More terms from Eric Rogoyski, May 15 1997

Name clarified by Andrew Howroyd, Sep 05 2018

A180260 Number of not necessarily connected 8-regular simple graphs on n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848874, 214755319657939505396, 187549729101764460261505, 186685399408147545744203915, 210977245260028917322933165888

Offset: 0

Jason Kimberley, Jan 17 2011

The Euler transformation currently does nothing: for n < 18, a(n) = A014378(n).

			The a(0)=1 graph is K_0 (vacuously 8-regular).
The a(9)=1 graph is K_9.

Eric Weisstein's World of Mathematics, Octic Graph

8-regular simple graphs: A014378 (connected), A165878 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A000012 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), this sequence (k=8).
8-regular not necessarily connected graphs: this sequence (simple graphs), A129437 (multigraphs with loops allowed), A129426 (multigraphs with loops forbidden).

A014378 = Cases[Import["https://oeis.org/A014378/b014378.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014378] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014378.

a(17)-a(22) from Andrew Howroyd, Mar 08 2020

A185643 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 4, 5, 3, 1, 1, 0, 0, 2, 0, 16, 0, 4, 0, 1, 0, 0, 2, 15, 58, 59, 21, 5, 1, 1, 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1, 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1, 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1

Offset: 1

Jason Kimberley, Feb 07 2013

			01: 0;
02: 0, 0;
03: 0, 0, 1;
04: 0, 0, 0, 1;
05: 0, 0, 0, 0, 1;
06: 0, 0, 1, 1, 1, 1;
07: 0, 0, 1, 0, 2, 0, 1;
08: 0, 0, 1, 4, 5, 3, 1, 1;
09: 0, 0, 2, 0, 16, 0, 4, 0, 1;
10: 0, 0, 2, 15, 58, 59, 21, 5, 1, 1;
11: 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1;
12: 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1;
13: 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1;
14: 0, 0, 6, 428, 87973, 3459379, 21609300, 21609300, 3459386, 88193, 540, 13, 1, 1;
15: 0, 0, 9, 0, 803973, 0, 1470293675, 0, 1470293676, 0, 805579, 0, 17, 0, 1;
16: 0, 0, 10, 3406, 8020967, 2585136353, 113314233804, 733351105934, 733351105934, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Jason Kimberley, Table of i, a(i)=E(n,k) for i = 1..136 (n = 1..16)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A198313(n).

Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), this sequence (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

E(n,k) = A186733(n,k) + A210703(n,k), noting that A210703 is a tabf.

E(n,k) = A051031(n,k) - A185304(n,k), noting that A185304 is a tabf.

A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 8, 12, 8, 4, 1, 1, 4, 10, 24, 24, 10, 4, 1, 1, 5, 17, 70, 118, 70, 17, 5, 1, 1, 5, 24, 172, 634, 634, 172, 24, 5, 1, 1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1, 1, 6, 50, 1530, 35500, 187990, 187990, 35500, 1530, 50, 6, 1

Offset: 0

Andrew Howroyd, Mar 11 2020

A half-edge is like a loop except it only adds 1 to the degree of its vertex.
T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333159.
T(n,k) is the number of simple graphs on n unlabeled vertices with every vertex degree being either k or k-1.

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  2,   1;
  1, 3,  3,   3,    1;
  1, 3,  4,   4,    3,    1;
  1, 4,  8,  12,    8,    4,    1;
  1, 4, 10,  24,   24,   10,    4,   1;
  1, 5, 17,  70,  118,   70,   17,   5,  1;
  1, 5, 24, 172,  634,  634,  172,  24,  5, 1;
  1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1;
  ...
The a(2,1) = 2 adjacency matrices are:
  [0 1]  [1 0]
  [1 0]  [0 1]
.
The A(4,2) = 3 adjacency matrices are:
  [0 0 1 1]   [1 1 0 0]   [1 1 0 0]
  [0 0 1 1]   [1 1 0 0]   [1 0 1 0]
  [1 1 0 0]   [0 0 1 1]   [0 1 0 1]
  [1 1 0 0]   [0 0 1 1]   [0 0 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..230

Columns k=0..3 are A000012, A004526(n+2), A186417, A333163.
Row sums are A333162.
Central coefficients are A333166.
Cf. A051031, A333157, A333159.

T(n,k) = T(n, n-k).

A350910 Triangle read by rows: T(n,k) is the number of k-regular digraphs on n unlabeled nodes, k = 0..n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 4, 23, 23, 4, 1, 1, 4, 92, 415, 92, 4, 1, 1, 7, 624, 19041, 19041, 624, 7, 1, 1, 8, 5021, 1104045, 6510087, 1104045, 5021, 8, 1, 1, 12, 47034, 79818336, 2983458766, 2983458766, 79818336, 47034, 12, 1

Offset: 1

Andrew Howroyd, Jan 29 2022

			Triangle begins:
  1;
  1, 1;
  1, 1,   1;
  1, 2,   2,     1;
  1, 2,   5,     2,     1;
  1, 4,  23,    23,     4,   1;
  1, 4,  92,   415,    92,   4, 1;
  1, 7, 624, 19041, 19041, 624, 7, 1;
  ...

Columns k=0..6 are A000012, A002865, A219889, A219890, A219891, A219892, A219893.
Row sums are A350911.
Cf. A051031 (graphs), A329228 (semi-regular), A350912.

A350912 Triangle read by rows: T(n,k) is the number of oriented graphs on n unlabeled nodes whose underlying graph is k-regular, k = 0..n-1.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 4, 4, 1, 0, 4, 0, 12, 1, 1, 12, 62, 112, 56, 1, 0, 18, 0, 1602, 0, 456, 1, 1, 40, 2062, 32263, 92980, 46791, 6880, 1, 0, 68, 0, 748576, 0, 11210264, 0, 191536, 1, 1, 140, 103827, 19349672, 616991524, 3319462470, 2729098064, 292115960, 9733056

Offset: 1

Andrew Howroyd, Jan 29 2022

The sum of the in-degree and out-degree at each node is k.

a(2*n,2*n-2) is the number of orientations (up to isomorphism) of the n-cocktail party graph. - Pontus von Brömssen, Apr 03 2022

			Triangle begins:
  1;
  1, 1;
  1, 0,  2;
  1, 1,  4,  4;
  1, 0,  4,  0,   12;
  1, 1, 12, 62,  112, 56;
  1, 0, 18,  0, 1602,  0, 456;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..253 (rows 1..22)

Row sums are A350913.
Main diagonal is A000568.
The labeled version is A351263.
Cf. A051031 (graphs), A350910 (digraphs).

cp's OEIS Frontend

A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A165626 Number of 5-regular graphs (quintic graphs) on 2n vertices.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A165627 Number of 6-regular graphs (sextic graphs) on n vertices.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A165628 Number of 7-regular graphs (septic graphs) on 2n vertices.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A008327 Triangle read by rows: T(n,k) is the number of simple regular bipartite graphs with 2n nodes and degree k, (0 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A180260 Number of not necessarily connected 8-regular simple graphs on n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185643 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs