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.

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A184951 Irregular triangle C(n,g) counting the connected 5-regular simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 3, 60, 1, 7848, 1, 3459383, 7, 2585136675, 388, 2807105250897, 406824
Offset: 3

Views

Author

Jason Kimberley, Jan 10 2012

Keywords

Comments

The first column is for girth at least 3. The row length sequence starts: 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4. The row length is incremented to g-2 when 2n reaches A054760(5,g).

Examples

			1;
3;
60, 1;
7848, 1;
3459383, 7;
2585136675, 388;
2807105250897, 406824;

Links

Crossrefs

Connected 5-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A006821 (g=3), A058275 (g=4), A205295 (g=5).
Connected 5-regular simple graphs with girth exactly g: A184950 (triangle); chosen g: A184953 (g=3), A184954 (g=4), A184955 (g=5).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: A185131 (k=3), A184941 (k=4), this sequence (k=5), A184961 (k=6), A184971 (k=7), A184981 (k=8).

Extensions

a(14) from Jason Kimberley, Dec 26 2012

A184960 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 4, 21, 266, 7848, 1, 367860, 0, 21609299, 1, 1470293674, 1, 113314233799, 9, 9799685588930, 6
Offset: 7

Views

Author

Jason Kimberley, Feb 24 2011

Keywords

Comments

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3. The row length is incremented to g-2 when n reaches A054760(6,g).

Examples

			Triangle begins:
1;
1;
4;
21;
266;
7848, 1;
367860, 0;
21609299, 1;
1470293674, 1;
113314233799, 9;
9799685588930, 6;
?, 267;
?, 3727;
?, 483012;
?, 69823723;
?, 14836130862;
The C(40,5)=1 (see the a-file) graph, the unique (6,5)-cage, is the complement of a Petersen graph inside the Hoffman-Singleton graph [from Brouwer link].
The first known of C(42,5)>=1 graph(s) has automorphism group of order 5040 and these adjacency lists:
1 : 2 3 4 5 6 7
2 : 1 8 9 10 11 12
3 : 1 13 14 15 16 17
4 : 1 18 19 20 21 22
5 : 1 23 24 25 26 27
6 : 1 28 29 30 31 32
7 : 1 33 34 35 36 37
8 : 2 13 18 23 28 38
9 : 2 14 19 24 33 39
10 : 2 15 20 29 34 40
11 : 2 16 25 30 35 41
12 : 2 21 26 31 36 42
13 : 3 8 21 27 34 41
14 : 3 9 26 28 37 40
15 : 3 10 22 25 31 39
16 : 3 11 19 32 36 38
17 : 3 20 23 30 33 42
18 : 4 8 25 32 33 40
19 : 4 9 16 27 29 42
20 : 4 10 17 26 35 38
21 : 4 12 13 30 37 39
22 : 4 15 24 28 36 41
23 : 5 8 17 29 36 39
24 : 5 9 22 30 34 38
25 : 5 11 15 18 37 42
26 : 5 12 14 20 32 41
27 : 5 13 19 31 35 40
28 : 6 8 14 22 35 42
29 : 6 10 19 23 37 41
30 : 6 11 17 21 24 40
31 : 6 12 15 27 33 38
32 : 6 16 18 26 34 39
33 : 7 9 17 18 31 41
34 : 7 10 13 24 32 42
35 : 7 11 20 27 28 39
36 : 7 12 16 22 23 40
37 : 7 14 21 25 29 38
38 : 8 16 20 24 31 37
39 : 9 15 21 23 32 35
40 : 10 14 18 27 30 36
41 : 11 13 22 26 29 33
42 : 12 17 19 25 28 34

Links

Crossrefs

Connected 6-regular simple graphs with girth at least g: A184961 (triangle); chosen g: A006822 (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184963 (g=3), A184964 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), this sequence (k=6), A184970 (k=7), A184980 (k=8).

Extensions

After approximately 390 processor days of computation with genreg, C(41,5)=0.

A184961 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth at least g.

Original entry on oeis.org

1, 1, 4, 21, 266, 7849, 1, 367860, 0, 21609300, 1, 1470293675, 1, 113314233808, 9, 9799685588936, 6
Offset: 7

Views

Author

Jason Kimberley, Jan 10 2012

Keywords

Comments

The first column is for girth at least 3. The row length sequence starts: 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3. The row length is incremented to g-2 when n reaches A054760(6,g).

Examples

			Triangle begins:
1;
1;
4;
21;
266;
7849, 1;
367860, 0;
21609300, 1;
1470293675, 1;
113314233808, 9;
9799685588936, 6;

Links

Crossrefs

Connected 6-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A006822 (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: A184960 (triangle); chosen g: A184963 (g=3), A184964 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: A185131 (k=3), A184941 (k=4), A184951 (k=5), this sequence (k=6), A184971 (k=7), A184981 (k=8).

A184970 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 5, 1547, 21609300, 1, 733351105933, 1
Offset: 4

Views

Author

Jason Kimberley, Feb 25 2011

Keywords

Comments

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2. The row length is incremented to g-2 when 2n reaches A054760(7,g).

Examples

			1;
5;
1547;
21609300, 1;
733351105933, 1;
?, 8;
?, 741;
?, 2887493;

Links

Crossrefs

Connected 7-regular simple graphs with girth at least g: A184971 (triangle); chosen g: A014377 (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184973 (g=3), A184974 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), A184960 (k=6), this sequence (k=7), A184980 (k=8).

A184971 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 5, 1547, 21609301, 1, 733351105934, 1
Offset: 4

Views

Author

Jason Kimberley, Jan 10 2012

Keywords

Comments

The first column is for girth at least 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2. The row length is incremented to g-2 when 2n reaches A054760(7,g).

Examples

			1;
5;
1547;
21609301, 1;
733351105934, 1;
?, 8;
?, 741;
?, 2887493;

Links

Crossrefs

Connected 7-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A014377 (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: A184970 (triangle); chosen g: A184973 (g=3), A184974 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: A185131 (k=3), A184941 (k=4), A184951 (k=5), A184961 (k=6), this sequence (k=7), A184981 (k=8).

A191595 Order of smallest n-regular graph of girth 5.

Original entry on oeis.org

5, 10, 19, 30, 40, 50
Offset: 2

Views

Author

N. J. A. Sloane, Jun 07 2011

Keywords

Comments

Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012
Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

Links

Crossrefs

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).
Moore lower bound on the orders of (k,g) cages: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306(k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10),A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 02 2011

Formula

a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011

Extensions

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013

A218553 Order of (5,n) cage, i.e., minimal order of 5-regular graph of girth n.

Original entry on oeis.org

6, 10, 30, 42
Offset: 3

Views

Author

Arkadiusz Wesolowski, Nov 02 2012

Keywords

Comments

a(7) <= 152, a(8) = 170, a(12) = 2730. - From Royle's page via Jason Kimberley, Dec 21 2012

Links

Crossrefs

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), this sequence (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).

Formula

a(n) >= A061547(n+1).

Extensions

a(7) deleted by Jason Kimberley, Dec 21 2012

A218554 Order of (6,n) cage, i.e., minimal order of 6-regular graph of girth n.

Original entry on oeis.org

7, 12, 40, 62
Offset: 3

Views

Author

Arkadiusz Wesolowski, Nov 02 2012

Keywords

Comments

a(7) <= 294, a(8) = 312, a(12) = 7812. - From Royle's page via Jason Kimberley, Dec 26 2012

Links

Crossrefs

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), this sequence (6,n), A218555 (7,n), A191595 (n,5).

Formula

a(n) >= A198306(n).

Extensions

a(7) deleted by Jason Kimberley, Dec 21 2012

A218555 Order of (7,n) cage, i.e., minimal order of 7-regular graph of girth n.

Original entry on oeis.org

8, 14, 50, 90
Offset: 3

Views

Author

Arkadiusz Wesolowski, Nov 02 2012

Keywords

Comments

a(8) <= 658, a(12) <= 32928. - Jason Kimberley, Dec 29 2012

Links

Crossrefs

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), this sequence (7,n), A191595 (n,5).

Formula

a(n) >= A198307(n).

Extensions

Edited by Jason Kimberley, Dec 21 2012

A184946 Number of connected 4-regular simple graphs on n vertices with girth exactly 6.

Original entry on oeis.org

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, 1, 0, 1, 0, 4, 0, 19, 0, 1272, 25, 494031, 13504
Offset: 0

Views

Author

Jason Kimberley, Feb 27 2011

Keywords

Comments

First differs from A058348 at n = A054760(4,7) = 67.

Crossrefs

Connected 4-regular simple graphs with girth at least g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5), this sequence (g=6).
Previous Showing 11-20 of 22 results. Next

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

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