A185245 -id:A185245 - cp's OEIS frontend

A033483 Number of disconnected 4-valent (or quartic) graphs with n nodes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 8, 25, 88, 378, 2026, 13351, 104595, 930586, 9124662, 96699987, 1095469608, 13175272208, 167460699184, 2241578965849, 31510542635443, 464047929509794, 7143991172244290, 114749135506381940, 1919658575933845129, 33393712487076999918, 603152722419661386031

Offset: 0

Ronald C. Read

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Quartic Graph

4-regular simple graphs: A006820 (connected), this sequence (disconnected), A033301 (not necessarily connected). - Jason Kimberley, Jan 08 2011
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), this sequence (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
Disconnected 4-regular simple graphs with girth at least g: this sequence (g=3), A185244 (g=4), A185245 (g=5), A185246 (g=6).

A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] - A006820 (* Jean-François Alcover, Jan 21 2020, updated Mar 17 2020 *)

a(n) = A033301(n) - A006820(n) = Euler_transformation(A006820) - A006820.

a(n) = A068933(n, 4). - Jason Kimberley, Sep 27 2009 and Jan 08 2011

Terms a(16)-a(18) from Martin Fuller, Dec 04 2006
Terms a(19)-a(26) from Jason Kimberley, Sep 27 2009 and Dec 30 2010
Terms a(27)-a(33), due to the extension of A006820 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020

A185244 Number of disconnected 4-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 15, 35, 247, 1692, 17409, 197924, 2492824, 33117880, 461597957, 6709514218, 101153412903, 1597440868898

Offset: 0

Jason Kimberley, Feb 22 2011

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

4-regular simple graphs with girth at least 4: A033886 (connected), this sequence (disconnected), A185344 (not necessarily connected).
Disconnected 4-regular simple graphs with girth at least g: A033483 (g=3), this sequence (g=4), A185245 (g=5), A185246 (g=6).
Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), this sequence (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

a(n) = A185344(n) - A033886(n) = Euler_transformation(A033886)(n) - A033886(n).

a(n) = A185044(n) + A185245(n).

a(31) appended by the author once A033886(23) was known, Nov 03 2011

a(31) corrected by the author, Jan 05 2013

A058343 Number of connected 4-regular simple graphs on n 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, 0, 0, 0, 0, 1, 2, 8, 131, 3917, 123859, 4131991, 132160608, 4018022149, 118369811960

Offset: 0

N. J. A. Sloane, Dec 17 2000

The null graph on 0 vertices is vacuously connected and 4-regular; since it is acyclic, it has infinite girth. [From Jason Kimberley, Jan 29 2011]

M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146. [From Jason Kimberley, Jan 29 2011]

Jason Kimberley, Connected regular graphs with girth at least 5
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs

Contribution from Jason Kimberley, 2010, 2011, and 2012: (Start)
4-regular simple graphs with girth at least 5: this sequence (connected), A185245 (disconnected), A185345 (not necessarily connected).
Connected k-regular simple graphs with girth at least 5: A186725 (all k), A186715 (triangle); A185115 (k=2), A014372 (k=3), this sequence (k=4), A205295 (k=5).
Connected 4-regular simple graphs with girth at least g: A006820 (g=3), A033886 (g=4), this sequence (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5). (End)

Terms a(27) and a(28) were appended by Jason Kimberley, from running Meringer's GENREG for 58 and 1563 processor days at U. Ncle, on Mar 19 and Jun 28 2010.

A185246 Number of disconnected 4-regular simple graphs on n vertices with girth at least 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, 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, 5, 0, 23, 0, 1301, 25, 495379, 13529

Offset: 0

Jason Kimberley, Feb 22 2011

Jason Kimberley, Disconnected regular graphs with girth at least 6
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

4-regular simple graphs with girth at least 4: A058348 (connected), this sequence (disconnected), A185346 (not necessarily connected).
Disconnected 4-regular simple graphs with girth at least g: A033483 (g=3), A185244 (g=4), A185245 (g=5), this sequence (g=6).
Disconnected k-regular simple graphs with girth at least 6: A185216 (all k), A185206 (triangle); A185226 (k=2), A185236 (k=3), this sequence (k=4).

a(n) = A185346(n) - A058348(n) = Euler_transformation(A058348)(n) - A058348(n).

A185044 Number of disconnected 4-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 15, 35, 247, 1692, 17409, 197924, 2492824, 33117880, 461597957, 6709514218, 101153412903, 1597440868898

Offset: 0

Jason Kimberley, Nov 04 2011

Only one component need have girth exactly four; the other components need only have girth at least four.

First differs from A185244 at n = 38, the smallest n where A185245 is nonzero.

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth exactly g

Disconnected 4-regular simple graphs with girth exactly g: A185043 (g=3), this sequence (g=4).

Disconnected k-regular simple graphs with girth exactly 4: A185034 (k=3), this sequence (k=4).

a(n) = A185244(n) - A185245(n).

a(n) = A185144(n) - A184944(n).

a(31) corrected by the author, propagated from A185244, Jan 05 2013

cp's OEIS Frontend

A033483 Number of disconnected 4-valent (or quartic) graphs with n nodes.

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A185244 Number of disconnected 4-regular simple graphs on n vertices with girth at least 4.

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A058343 Number of connected 4-regular simple graphs on n vertices with girth at least 5.

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A185246 Number of disconnected 4-regular simple graphs on n vertices with girth at least 6.

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A185044 Number of disconnected 4-regular simple graphs on n vertices with girth exactly 4.

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