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 1-10 of 11 results. Next

A263341 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n vertices with independence number k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 13, 15, 4, 1, 1, 37, 82, 30, 5, 1, 1, 106, 578, 301, 51, 6, 1, 1, 409, 6021, 4985, 842, 80, 7, 1, 1, 1896, 101267, 142276, 27107, 1995, 117, 8, 1, 1, 12171, 2882460, 7269487, 1724440, 112225, 4210, 164, 9, 1, 1, 105070, 138787233, 655015612, 210799447, 13893557, 388547, 8165, 221, 10, 1
Offset: 1

Views

Author

Christian Stump, Oct 15 2015

Keywords

Comments

The independence number of a graph is the maximum size of an independent set.
Row sums give A000088, n >= 1.
T(n,k) is also the number of graphs on n vertices such that a largest clique is of size k. - Geoffrey Critzer, Sep 23 2016
T(n,k) is also the number of graphs on n vertices such that the size of a smallest vertex cover is n-k. - Geoffrey Critzer, Sep 23 2016
T(n,k) is also the number of graphs on n vertices with independence number k. - Eric W. Weisstein, May 17 2017
For any graph the independence number is greater than or equal to the independent domination number (A332402) and less than or equal to the upper domination number (A332403). - Andrew Howroyd, Feb 19 2020

Examples

			Triangle begins:
  1;
  1,     1;
  1,     2,       1;
  1,     6,       3,       1;
  1,    13,      15,       4,       1;
  1,    37,      82,      30,       5,      1;
  1,   106,     578,     301,      51,      6,    1;
  1,   409,    6021,    4985,     842,     80,    7,   1;
  1,  1896,  101267,  142276,   27107,   1995,  117,   8, 1;
  1, 12171, 2882460, 7269487, 1724440, 112225, 4210, 164, 9, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Columns 2..9 are A052450, A052451, A052452, A077392, A077393, A077394, A205577, A205578.
Transpose of A287024.
Cf. A115196, A126744 (clique number of connected graphs), A294490 (independence number of connected graphs).
Cf. A263284, A332402, A332403.

Extensions

a(21)-a(28) from Geoffrey Critzer, Sep 22 2016
Rows 8-10 from Eric W. Weisstein, May 16 2017
Rows 11-13 from Brendan McKay, Feb 18 2020
Name clarified by Andrew Howroyd, Feb 18 2020

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

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

A115196 Triangle read by rows formed from nonzero entries in table of number of graphs on n nodes with clique number k.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 15, 13, 1, 5, 30, 82, 37, 1, 6, 51, 301, 578, 106, 1, 7, 80, 842, 4985, 6021, 409, 1, 8, 117, 1995, 27107, 142276, 101267, 1896, 1, 9, 164, 4210, 112225, 1724440, 7269487, 2882460, 12171
Offset: 2

Views

Author

N. J. A. Sloane, based on email from Keith Briggs, Apr 03 2006

Keywords

Examples

			Table: number of graphs on n nodes with clique number k
n = .1...2...3...4....5....6.....7......8........9.......10.
k ----------------------------------------------------------
2....0...1...2...6...13...37...106....409.....1896....12171 = A052450
3....0...0...1...3...15...82...578...6021...101267..2882460 = A052451
4....0...0...0...1...4....30...301...4985...142276..7269487 = A052452
5....0...0...0...0...1....5.....51....842....27107..1724440 = A077392
6....0...0...0...0...0....1......6.....80.....1995...112225 = A077393
7....0...0...0...0...0....0......1......7......117.....4210 = A077394
8....0...0...0...0...0....0......0......1........8......164 = A205577
9....0...0...0...0...0....0......0......0........1........9 = A205578
10...0...0...0...0...0....0......0......0........0........1.

Links

Crossrefs

Cf. A287024, A263341. Partial column sums: A304124, A304125.

Formula

1+Sum_{k>=2} T(n,k) = A000088(n). - R. J. Mathar, May 06 2018

A304124 Number of simple graphs with n vertices which contain no K4 subgraph.

Original entry on oeis.org

1, 2, 4, 10, 29, 120, 685, 6431, 103164, 2894632, 138892304, 11118977705, 1459412127955
Offset: 1

Views

Author

Brendan McKay, May 06 2018

Keywords

Comments

The graphs do not need to be connected.

Crossrefs

Cf. A000088, A006785 (no K3), A115196 (graphs by clique number), A304125 (no K5).

Formula

a(n) = 1+A052450(n)+A052451(n).

Extensions

a(13) from Brendan McKay, May 08 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

A304125 Number of simple graphs with n vertices which contain no K5 subgraph.

Original entry on oeis.org

1, 2, 4, 11, 33, 150, 986, 11416, 245440, 10164119, 793907916, 114150623175, 29709609171994
Offset: 1

Views

Author

Brendan McKay, May 06 2018

Keywords

Comments

The graphs do not need to be connected.

Crossrefs

Cf. A000088, A006785 (no K3), A115196 (graphs by clique number), A304124 (no K4).

Formula

a(n) = 1+A052450(n)+A052451(n)+A052452(n).

Extensions

a(13) from Brendan McKay, May 08 2018
Showing 1-10 of 11 results. Next

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

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