A006799 -id:A006799 - cp's OEIS frontend

A319367 Triangle read by rows: T(n,k) is the number of simple vertex transitive graphs with n nodes and valency k, (0 <= k < n).

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 0, 3, 0, 2, 0, 1, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 4, 7, 11, 13, 13, 11, 7, 4, 1, 1, 1, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1

Offset: 1

Andrew Howroyd, Sep 17 2018

			Triangle begins, n >= 1, 0 <= k < n:
  1;
  1, 1;
  1, 0, 1;
  1, 1, 1, 1;
  1, 0, 1, 0,  1;
  1, 1, 2, 2,  1,  1;
  1, 0, 1, 0,  1,  0,  1;
  1, 1, 2, 3,  3,  2,  1,  1;
  1, 0, 2, 0,  3,  0,  2,  0,  1;
  1, 1, 2, 3,  4,  4,  3,  2,  1,  1;
  1, 0, 1, 0,  2,  0,  2,  0,  1,  0,  1;
  1, 1, 4, 7, 11, 13, 13, 11,  7,  4,  1,  1;
  1, 0, 1, 0,  3,  0,  4,  0,  3,  0,  1,  0,  1;
  1, 1, 2, 3,  6,  6,  9,  9,  6,  6,  3,  2,  1,  1;
  1, 0, 3, 0,  8,  0, 12,  0, 12,  0,  8,  0,  3,  0, 1;
  1, 1, 3, 7, 16, 27, 40, 48, 48, 40, 27, 16,  7,  3, 1, 1;
  1, 0, 1, 0,  4,  0,  7,  0, 10,  0,  7,  0,  4,  0, 1, 0, 1;
  1, 1, 4, 7, 16, 24, 38, 45, 54, 54, 45, 38, 24, 16, 7, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..496
B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176. (Annotated scanned copy)
Gordon Royle, Transitive Graphs.
Eric Weisstein's World of Mathematics, Vertex-Transitive Graph.

Columns k=2..12 (even n only for odd k) are A023645, A023646, A023647, A023640, A023641, A023642, A023643, A023644, A023637, A023638, A023639.
Row sums are A006799.
Cf. A319368, A319372.

A185959 Number of Cayley graphs on n nodes.

Original entry on oeis.org

1, 2, 2, 4, 3, 8, 4, 14, 9, 20, 8, 74, 14, 56, 44, 278, 36, 376, 60, 1132, 240, 816, 188, 15394, 464, 4104, 1434, 25784, 1182, 45184, 2192, 659232, 6768, 131660, 11144, 1959040, 14602, 814216, 48462, 13055904, 52488, 9461984, 99880, 39134544, 399126, 34333800, 364724

Offset: 1

Eric W. Weisstein, Feb 07 2011

First differs from A006799 at n = 10.

Derek Holt and Gordon Royle, A Census of Small Transitive Groups and Vertex-Transitive Graphs, arXiv:1811.09015 [math.CO], 2018.
G. Royle, Transitive graphs
Eric Weisstein's World of Mathematics, Cayley Graph

Row sums of A319372.

Cf. A006792, A006799.

A006792 = Join[Array[0&, 10], Cases[Import["https://oeis.org/A006792/b006792.txt", "Table"], {, }][[All, 2]]];
A006799 = Cases[Import["https://oeis.org/A006799/b006799.txt", "Table"], {, }][[All, 2]];
a[n_] := A006799[[n]] - A006792[[n+1]];
Array[a, 47] (* Jean-François Alcover, Sep 05 2019 *)

a(n) = A006799(n) - A006792(n).

a(32)-a(47) from Danny Rorabaugh, Nov 26 2018

A006792 Number of n-node vertex-transitive graphs which are not Cayley graphs.

Original entry on oeis.org

2, 0, 0, 0, 0, 4, 8, 0, 4, 0, 82, 0, 0, 0, 112, 0, 132, 0, 66, 0, 1124, 0, 18170, 0, 920, 6, 4162, 0, 0, 0, 48266, 0, 242, 0, 96, 294, 0, 0

Offset: 10

N. J. A. Sloane

McKay, Brendan D.; Royle, Gordon F.; The transitive graphs with at most 26 vertices. Ars Combin. 30 (1990), 161-176.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. McKay, Email to N. J. A. Sloane, Jul. 1991
Brendan D. McKay, Cheryl E. Praeger, Vertex-transitive graphs which are not Cayley graphs, I. J. Austral. Math. Soc. Ser. A 56 (1994), no. 1, 53-63.
G. Royle, Transitive graphs
Steven Skiena, A Database of Graphs in Combinatorica Format.
Eric Weisstein's World of Mathematics, Noncayley Graph

Cf. A006799, A185959.

a(n) = A006799(n) - A185959(n). - Andrew Howroyd, Nov 27 2018

More terms from Vladeta Jovovic, Jun 30 2007
a(32)-a(47) from Andrew Howroyd, Nov 27 2018
Duplicate a(32) removed by Andrew Howroyd, Sep 05 2019

A319368 Triangle read by rows: T(n,k) is the number of simple connected vertex transitive graphs with n nodes and valency k, (0 <= k < n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 3, 2, 1, 1, 0, 0, 1, 0, 3, 0, 2, 0, 1, 0, 0, 1, 3, 3, 4, 3, 2, 1, 1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 0, 1, 4, 10, 12, 13, 11, 7, 4, 1, 1, 0, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1

Offset: 1

Andrew Howroyd, Sep 17 2018

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 1, 0,  1;
  0, 0, 1, 2,  1,  1;
  0, 0, 1, 0,  1,  0,  1;
  0, 0, 1, 2,  3,  2,  1,  1;
  0, 0, 1, 0,  3,  0,  2,  0,  1;
  0, 0, 1, 3,  3,  4,  3,  2,  1,  1;
  0, 0, 1, 0,  2,  0,  2,  0,  1,  0,  1;
  0, 0, 1, 4, 10, 12, 13, 11,  7,  4,  1,  1;
  0, 0, 1, 0,  3,  0,  4,  0,  3,  0,  1,  0,  1;
  0, 0, 1, 3,  5,  6,  8,  9,  6,  6,  3,  2,  1,  1;
  0, 0, 1, 0,  7,  0, 12,  0, 12,  0,  8,  0,  3,  0, 1;
  0, 0, 1, 4, 13, 25, 39, 47, 48, 40, 27, 16,  7,  3, 1, 1;
  0, 0, 1, 0,  4,  0,  7,  0, 10,  0,  7,  0,  4,  0, 1, 0, 1;
  0, 0, 1, 5, 12, 23, 36, 45, 53, 54, 45, 38, 24, 16, 7, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..496
B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176. (Annotated scanned copy)
Gordon Royle, Transitive Graphs
Eric Weisstein's World of Mathematics, Vertex-Transitive Graph.

Row sums are A006800.

Cf. A319367.

T(n,k) = Sum_{d|n} moebius(n/d) * A319367(d,k).

A006800 Number of connected vertex-transitive graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 3, 10, 7, 18, 7, 64, 13, 51, 44, 272, 35, 365, 59, 1190, 235, 807, 187, 15422, 461, 4221, 1425, 25792, 1181, 46236, 2191, 677116, 6759, 132543, 11144, 1962756, 14601, 814155, 48447, 13102946, 52487, 9461929, 99879, 39133822, 399365, 34333611, 364723

Offset: 1

N. J. A. Sloane

B. D. McKay, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176. (Annotated scanned copy)
B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176.
G. Royle, Transitive graphs
Steven Skiena, A Database of Graphs in Combinatorica Format.
Eric Weisstein's World of Mathematics, Vertex-Transitive Graph

Row sums of A319368.

Cf A006799.

A006799 = Cases[Import["https://oeis.org/A006799/b006799.txt", "Table"], {, }][[All, 2]];
Table[Sum[MoebiusMu[n/d] A006799[[d]], {d, Divisors[n]}], {n, 1, Length[A006799]}] (* Jean-François Alcover, Aug 20 2019 *)

Moebius transform of A006799. - Andrew Howroyd, Sep 18 2018

More terms from Vladeta Jovovic, Jun 30 2007

a(32)-a(47) from Andrew Howroyd, Nov 27 2018

A330697 Numbers k such that every vertex-transitive graph of order k is a Cayley-graph.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 17, 19, 21, 22, 23, 25, 27, 29, 31, 33, 37, 38, 39, 41, 43, 46, 47, 49, 53, 55, 59, 61, 62, 67, 69, 71, 73, 79, 83, 86, 87, 89, 93, 94, 95, 97, 101, 103, 107, 109, 113, 115, 118, 121, 123, 125, 127, 129, 131, 133, 134, 137, 139

Offset: 1

M. Farrokhi D. G., Dec 26 2019

nonn

Numbers k for which A185959(k) = A006799(k).
Numbers k for which A006792(k) = 0.
In the Farrokhi reference, a Cayley number is a number k such that all vertex transitive graphs of order k are Cayley graphs.

M. Farrokhi D. G., List of known Cayley numbers less than or equal to 1000000
M. Farrokhi D. G., Cayley numbers
M. Farrokhi D. G., Cayley numbers (GitHub)

Cf. A006792, A006799, A185959, A330698.

# See M. Farrokhi Github repository for a GAP program.

A330698 Numbers k such that there exists a non-Cayley vertex-transitive graph of order k.

Original entry on oeis.org

10, 15, 16, 18, 20, 24, 26, 28, 30, 32, 34, 35, 36, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 88, 90, 91, 92, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 119, 120, 122, 124

Offset: 1

M. Farrokhi D. G., Dec 26 2019

nonn

Numbers k for which A185959(k) < A006799(k).
Numbers k for which A006792(k) <> 0.
In the Farrokhi reference, a Cayley number is a number k such that all vertex transitive graphs of order k are Cayley graphs.

M. Farrokhi D. G., List of known non-Cayley numbers less than or equal to 1000000
M. Farrokhi D. G., Cayley numbers
M. Farrokhi D. G., Cayley numbers (GitHub)

Cf. A006792, A006799, A185959, A330697.

A054917 Number of connected unlabeled vertex-transitive graphs with n nodes such that complement is also connected.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 2, 6, 5, 14, 6, 54, 12, 46, 40, 258, 34, 350, 58, 1166, 230, 798, 186, 15338, 458, 4206, 1416, 25734, 1180, 46164, 2190, 676830, 6750, 132506, 11138, 1962310, 14600, 814094, 48432, 13101722, 52486, 9461632, 99878, 39133004, 399310, 34333422, 364722

Offset: 1

N. J. A. Sloane, May 24 2000

nonn

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A006799, A006800.

nmax = 47;
A006799 = Cases[Import["https://oeis.org/A006799/b006799.txt", "Table"], {, }][[All, 2]];
A006800 = Table[Sum[MoebiusMu[n/d] A006799[[d]], {d, Divisors[n]}], {n, 1, nmax}];
a[n_] := 2*A006800[[n]] - A006799[[n]];
Array[a, nmax] (* Jean-François Alcover, Aug 27 2019, after Andrew Howroyd *)

a(n) = 2*A006800(n) - A006799(n).

Missing a(1) inserted and a(32)-a(47) from Andrew Howroyd, Nov 27 2018

A327754 Number of connected Cayley graphs on n nodes.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 3, 10, 7, 16, 7, 64, 13, 51, 40, 264, 35, 361, 59, 1110, 235, 807, 187, 15310, 461, 4089, 1425, 25726, 1181, 45118, 2191

Offset: 1

M. Farrokhi D. G., Sep 24 2019

Gordon Royle, Transitive Graphs

Cf. A185959, A006799, A006792.

cp's OEIS Frontend

A319367 Triangle read by rows: T(n,k) is the number of simple vertex transitive graphs with n nodes and valency k, (0 <= k < n).

Views

Author

Keywords

Examples

Links

Crossrefs

A185959 Number of Cayley graphs on n nodes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A006792 Number of n-node vertex-transitive graphs which are not Cayley graphs.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A319368 Triangle read by rows: T(n,k) is the number of simple connected vertex transitive graphs with n nodes and valency k, (0 <= k < n).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A006800 Number of connected vertex-transitive graphs with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A330697 Numbers k such that every vertex-transitive graph of order k is a Cayley-graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

A330698 Numbers k such that there exists a non-Cayley vertex-transitive graph of order k.

Views

Author

Keywords

Comments

Links

Crossrefs

A054917 Number of connected unlabeled vertex-transitive graphs with n nodes such that complement is also connected.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A327754 Number of connected Cayley graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs