cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A000066 Smallest number of vertices in trivalent graph with girth (shortest cycle) = n.

Original entry on oeis.org

4, 6, 10, 14, 24, 30, 58, 70, 112, 126
Offset: 3

Views

Author

N. J. A. Sloane

Keywords

Comments

Also called the order of the (3,n) cage graph.
Recently (unpublished) McKay and Myrvold proved that the minimal graph on 112 vertices is unique. - May 20 2003
If there are n vertices and e edges, then 3n=2e, so n is always even.
Current lower bounds for a(13)..a(32) are 202, 258, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072. - from Table 3 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012
Current upper bounds for a(13)..a(32) are 272, 384, 620, 960, 2176, 2560, 4324, 5376, 16028, 16206, 49326, 49608, 108906, 109200, 285852, 415104, 1141484, 1143408, 3649794, 3650304. - from Table 3 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

References

  • A. T. Balaban, Trivalent graphs of girth nine and eleven and relationships among cages, Rev. Roum. Math. Pures et Appl. 18 (1973) 1033-1043.
  • Brendan McKay, personal communication.
  • H. Sachs, On regular graphs with given girth, pp. 91-97 of M. Fiedler, editor, Theory of Graphs and Its Applications: Proceedings of the Symposium, Smolenice, Czechoslovakia, 1963. Academic Press, NY, 1964.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006787, A052453 (number of such graphs).
Orders of cages: A054760 (n,k), this sequence (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).

Formula

For all g > 2, a(g) >= A027383(g-1), with equality if and only if g = 3, 4, 5, 6, 8, or 12. - Jason Kimberley, Dec 21 2012 and Jan 01 2013

Extensions

Additional comments from Matthew Cook, May 15 2003
Balaban proved 112 as an upper bound for a(11). The proof that it is also a lower bound is in the paper by Brendan McKay, W. Myrvold and J. Nadon.

A052450 Number of n-node simple graphs having clique number 2.

Original entry on oeis.org

0, 1, 2, 6, 13, 37, 106, 409, 1896, 12171, 105070, 1262179, 20797001, 467871368, 14232552451, 581460254000, 31720840164949
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A052451, A052452, A052453, A077392, A077393, A077394.
a(n) = A006785(n) - 1. - Falk Hüffner, Nov 20 2015

Extensions

2 more terms from Eric W. Weisstein, Nov 03 2002
a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
More terms added using A006785 by Falk Hüffner, Nov 20 2015

A052451 Number of n-node simple graphs having clique number 3.

Original entry on oeis.org

0, 0, 1, 3, 15, 82, 578, 6021, 101267, 2882460, 138787233, 11117715525, 1459391330953
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Also, number of n-node simple graphs having independence number 3. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 3 of A263341.
Cf. A052450, A052452, A052453, A077392, A077393, A077394.

Extensions

2 more terms from Eric W. Weisstein, Nov 03 2002
a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A052452 Number of n-node simple graphs having clique number 4.

Original entry on oeis.org

0, 0, 0, 1, 4, 30, 301, 4985, 142276, 7269487, 655015612, 103031645470, 28250197044039
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Also, number of n-node simple graphs having independence number 4. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 4 of A263341.
Cf. A052450, A052451, A052453, A077392, A077393, A077394.

Extensions

2 more terms from Eric W. Weisstein, Nov 03 2002
a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A077392 Number of n-node simple graphs having clique number 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 51, 842, 27107, 1724440, 210799447, 47337500562, 19053225506745
Offset: 1

Views

Author

Eric W. Weisstein, Nov 03 2002

Keywords

Comments

Also, number of n-node simple graphs having independence number 5. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 5 of A263341.
Cf. A052450, A052451, A052452, A052453, A077393, A077394.

Extensions

a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A077393 Number of n-node simple graphs having clique number 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 80, 1995, 112225, 13893557, 3514580130, 1696127391214
Offset: 1

Views

Author

Eric W. Weisstein, Nov 03 2002

Keywords

Comments

Also, number of n-node simple graphs having independence number 6. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 6 of A263341.
Cf. A052450, A052451, A052452, A052453, A077392, A077394.

Extensions

a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A077394 Number of n-node simple graphs having clique number 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 7, 117, 4210, 388547, 87269430, 42603563082
Offset: 1

Views

Author

Eric W. Weisstein, Nov 03 2002

Keywords

Comments

Also, number of n-node simple graphs having independence number 7. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 7 of A263341.
Cf. A052450, A052451, A052452, A052453, A077392, A077393.

Extensions

a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A205577 Number of n-node simple graphs having clique number 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 8, 164, 8165, 1184155, 462435257
Offset: 1

Views

Author

Michael Sollami, Jan 29 2012

Keywords

Comments

Also, number of n-node simple graphs having independence number 8. - Andrew Howroyd, Oct 31 2017

Crossrefs

Column 8 of A263341.
Cf. A052450, A052451, A052452, A052453, A077392, A077393.

Extensions

a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A210709 Number of trivalent connected simple graphs with 2n nodes and girth at least 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18
Offset: 0

Views

Author

Jason Kimberley, Dec 20 2012

Keywords

Links

Crossrefs

Trivalent simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8), this sequence (g=9).
Trivalent simple graphs with girth exactly g: A198303 (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).

Formula

a(29) = a(A000066(9)/2) = A052453(9) = 18 is the number of (3,9) cages.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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