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.

A006821 Number of connected regular graphs of degree 5 (or quintic graphs) with 2n nodes.

Original entry on oeis.org

1, 0, 0, 1, 3, 60, 7848, 3459383, 2585136675, 2807105250897, 4221456117363365, 8516994770090547979, 22470883218081146186209, 75883288444204588922998674, 322040154704144697047052726990
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(0)=1 because the null graph (with no vertices) is vacuously 5-regular and connected.

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 648.
  • I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.
  • R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
5-regular simple graphs: this sequence (connected), A165655 (disconnected), A165626 (not necessarily connected).
Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), this sequence (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 5-regular simple graphs with girth at least g: this sequence (g=3), A058275 (g=4), A205295 (g=5).
Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), A184954 (g=4), A184955 (g=5).
Connected 5-regular graphs: A129430 (loops and multiple edges allowed), A129419 (no loops but multiple edges allowed), this sequence (no loops nor multiple edges). (End)

Formula

a(n) = A184953(n) + A058275(n).
a(n) = A165626(n) - A165655(n).
Inverse Euler transform of A165626.

Extensions

By running M. Meringer's GENREG for about 2 processor years at U. Newcastle, a(9) was found by Jason Kimberley, Nov 24 2009
a(10)-a(14) from Andrew Howroyd, Mar 10 2020

A058275 Number of connected 5-regular simple graphs on 2*n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 7, 388, 406824, 1125022325, 3813549359274
Offset: 0

Views

Author

N. J. A. Sloane, Dec 17 2000

Keywords

Comments

The null graph on 0 vertices is vacuously connected and 5-regular; since it is acyclic, it has infinite girth. - Jason Kimberley, Jan 30 2011

References

  • M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146. - Jason Kimberley, Jan 30 2011

Links

Crossrefs

From Jason Kimberley, Jan 30 and Nov 04 2011: (Start)
5-regular simple graphs on 2n vertices with girth at least 4: this sequence (connected), A185254 (disconnected), A185354 (not necessarily connected).
Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), A014371 (k=3), A033886 (k=4), this sequence (k=5), A058276 (k=6), A181153 (k=7), A181154 (k=8), A181170 (k=9).
Connected 5-regular simple graphs with girth at least g: A006821 (g=3), this sequence (g=4), A205295 (g=5).
Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), A184954 (g=4), A184955 (g=5). (End)

Formula

a(n) = A185354(n) - A185254(n);
This sequence is the inverse Euler transformation of A185354. - Jason Kimberley, Nov 04 2011

Extensions

Terms a(10) and a(11) appended, from running Meringer's GENREG for 3.8 and 7886 processor days at U. Ncle., by Jason Kimberley on Jun 28 2010

A006925 Number of connected trivalent graphs with 2n nodes and girth exactly 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 8, 48, 450, 5751, 90553, 1612905, 31297357, 652159389, 14499780660, 342646718608
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 647.
  • Gordon Royle, personal communication.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 5: this sequence (k=3), A184945 (k=4), A184955 (k=5).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); specified g: A006923 (g=3), A006924 (g=4), this sequence
(g=5), A006926 (g=6), A006927 (g=7).
Connected 3-regular simple graphs with girth at least g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

Formula

a(n) = A014372(n) - A014374(n).

Extensions

Definition corrected to include "connected", and "girth at least 5" minus "girth at least 6" formula provided by Jason Kimberley, Dec 12 2009

A184945 Number of connected 4-regular simple graphs on n vertices with girth exactly 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 8, 131, 3917, 123859, 4131991, 132160607, 4018022149, 118369811959
Offset: 0

Views

Author

Jason Kimberley, Feb 14 2011

Keywords

Examples

			a(0)=0 because even though the null graph (on zero vertices) is vacuously 4-regular and connected, since it is acyclic, it has infinite girth.
The a(19)=1 graph is the unique (4,5) cage: the Robertson graph (see also A159191). It has the following adjacency lists.
01 : 02 03 04 05
02 : 01 06 07 08
03 : 01 09 10 11
04 : 01 12 13 14
05 : 01 15 16 17
06 : 02 09 12 15
07 : 02 10 13 16
08 : 02 11 14 17
09 : 03 06 13 17
10 : 03 07 14 18
11 : 03 08 16 19
12 : 04 06 16 18
13 : 04 07 09 19
14 : 04 08 10 15
15 : 05 06 14 19
16 : 05 07 11 12
17 : 05 08 09 18
18 : 10 12 17 19
19 : 11 13 15 18

Links

Crossrefs

4-regular simple graphs with girth exactly 5: this sequence (connected), A185045 (disconnected), A185145 (not necessarily connected).
Connected k-regular simple graphs with girth exactly 5: A006925 (k=3), this sequence (k=4), A184955 (k=5).
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), this sequence (g=5).

Formula

a(n) = A058343(n) - A058348(n).

A184953 Number of connected 5-regular (or quintic) simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 3, 59, 7847, 3459376, 2585136287, 2807104844073
Offset: 0

Views

Author

Jason Kimberley, Feb 27 2011

Keywords

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 3: A006923 (k=3), A184943 (k=4), this sequence (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).
Connected 5-regular simple graphs with girth at least g: A006821 (g=3), A058275 (g=4).
Connected 5-regular simple graphs with girth exactly g: this sequence (g=3), A184954 (g=4), A184955 (g=5).

Formula

a(n) = A006821(n) - A058275(n).

A184954 Number of connected 5-regular simple graphs on 2n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 7, 388, 406824, 1125022325, 3813549359274
Offset: 0

Views

Author

Jason Kimberley, Feb 27 2011

Keywords

Links

Crossrefs

Connected k-regular simple graphs with girth exactly 4: A006924 (k=3), A184944 (k=4), this sequence (k=5), A184964 (k=6), A184974 (k=7).
Connected 5-regular simple graphs with girth at least g: A006821 (g=3), A058275 (g=4).
Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), this sequence (g=4), A184955 (g=5).

A205295 Number of connected 5-regular simple graphs on 2n vertices with girth at least 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 90
Offset: 0

Views

Author

Jason Kimberley, Jan 25 2012

Keywords

References

  • M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146.

Links

Crossrefs

Connected k-regular simple graphs with girth at least 5: A185115 (k=2), A014372 (k=3), A058343 (k=4), this sequence (k=5).
Connected 5-regular simple graphs with girth at least g: A006821 (g=3), A058275 (g=4), this sequence (g=5).
Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), A184954 (g=4), A184955 (g=5).

A184950 Irregular triangle C(n,g) counting the connected 5-regular simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 3, 59, 1, 7847, 1, 3459376, 7, 2585136287, 388, 2807104844073, 406824
Offset: 3

Views

Author

Jason Kimberley, Feb 24 2011

Keywords

Comments

The first column is for girth exactly 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;
59, 1;
7847, 1;
3459376, 7;
2585136287, 388;
2807104844073, 406824;
?, 1125022325;
?, 3813549359274;

Links

Crossrefs

Connected 5-regular simple graphs with girth at least g: A184951 (triangle); chosen g: A006821 (g=3), A058275 (g=4).
Connected 5-regular simple graphs with girth exactly g: this sequence (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 exactly g: A198303 (k=3), A184940 (k=4), this sequence (k=5), A184960 (k=6), A184970 (k=7), A184980 (k=8).

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
Showing 1-9 of 9 results.

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

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