A185959 -id:A185959 - cp's OEIS frontend

A006799 Number of vertex-transitive graphs with n nodes.

1, 2, 2, 4, 3, 8, 4, 14, 9, 22, 8, 74, 14, 56, 48, 286, 36, 380, 60, 1214, 240, 816, 188, 15506, 464, 4236, 1434, 25850, 1182, 46308, 2192, 677402, 6768, 132580, 11150, 1963202, 14602, 814216, 48462, 13104170, 52488, 9462226, 99880, 39134640, 399420, 34333800, 364724

Offset: 1

N. J. A. Sloane

CRC Handbook of Combinatorial Designs, 1996, p. 649.
Brendan McKay, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nino Bašić, Martin Knor, and Riste Škrekovski, On regular graphs with Šoltés vertices, arXiv:2303.11996 [math.CO], 2023.
Derek Holt and Gordon Royle, A Census of Small Transitive Groups and Vertex-Transitive Graphs, arXiv:1811.09015 [math.CO], 2018.
B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176. (Annotated scanned copy)
Brendan D. McKay, Gordon 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
Eric Weisstein's World of Mathematics, Cayley Graph

Row sums of A319367.

Cf. A006792, A006793, A006800, A185959.

Inverse Moebius transform of A006800. - Andrew Howroyd, Sep 18 2018

More terms from Vladeta Jovovic, Jun 30 2007

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

A319372 Triangle read by rows: T(n,k) is the number of Cayley graphs with n nodes and valency k, (0 <= k < n).

Original entry on oeis.org

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, 2, 4, 4, 2, 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

First differs from A319367 in row 10.

			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, 2,  4,  4,  2,  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,  7,  0, 11,  0, 11,  0,  7,  0,  3,  0, 1;
  1, 1, 3, 7, 15, 26, 39, 47, 47, 39, 26, 15,  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, 23, 38, 45, 53, 53, 45, 38, 23, 16, 7, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..496
Gordon Royle, Transitive Graphs
Eric Weisstein's World of Mathematics, Cayley Graph
Wikipedia, Cayley graph

Column k=3 is aerated A319374.
Row sums are A185959.
Cf. A319367, A319368.

A241164 Number of 2n-vertex connected cubic vertex-transitive graphs which are Cayley graphs.

Original entry on oeis.org

1, 2, 2, 2, 4, 3, 4, 5, 5, 3, 11, 4, 5, 7, 10, 4, 12, 5, 10, 10, 7, 5, 32, 8, 9, 13, 13, 6, 30, 7, 26, 11, 11, 11, 36, 8, 11, 14, 29, 8, 27, 9, 16, 18, 13, 9, 90, 13, 23, 15, 20, 10, 41, 19, 35, 18, 17, 11, 100, 12, 17, 26, 82, 17, 35, 13

Offset: 2

N. J. A. Sloane, Apr 19 2014

nonn

N. J. A. Sloane, Table of n, a(n) for n = 2..640, based on the work of Primož Potočnik, Pablo Spiga and Gabriel Verret
Primož Potočnik, Pablo Spiga and Gabriel Verret, A census of small connected cubic vertex-transitive graphs (see the sub-page Table.html)
G. Royle, Cubic transitive graphs

Cf. A032355, A185959.

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.

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

A006799 Number of vertex-transitive graphs with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A319372 Triangle read by rows: T(n,k) is the number of Cayley graphs with n nodes and valency k, (0 <= k < n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A241164 Number of 2n-vertex connected cubic vertex-transitive graphs which are Cayley graphs.

Views

Author

Keywords

Links

Crossrefs

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

A327754 Number of connected Cayley graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs