A185244 -id:A185244 - cp's OEIS frontend

A033886 Number of connected 4-regular simple graphs on n vertices with girth at least 4.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 12, 31, 220, 1606, 16828, 193900, 2452818, 32670330, 456028474, 6636066099, 100135577747, 1582718912968

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. - Jason Kimberley, Jan 29 2011

Jason Kimberley, Connected regular graphs with girth at least 4.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g.
Markus Meringer, Fast Generation of Regular Graphs and Construction of Cages, Journal of Graph Theory, Vol. 30, No. 2 (1999), 137-146.
Markus Meringer, Tables of Regular Graphs.

From Jason Kimberley, Mar 19 2010 and Jan 28 2011: (Start)
4-regular simple graphs with girth at least 4: this sequence (connected), A185244 (disconnected), A185344 (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), this sequence (k=4), A058275 (k=5), A058276 (k=6), A181153 (k=7), A181154 (k=8), A181170 (k=9).
Connected 4-regular simple graphs with girth at least g: A006820 (g=3), this sequence (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). (End)

By running M. Meringer's GENREG at U. Newcastle for 6.25, 107 and 1548 processor days, a(21), a(22), and a(23) were completed by Jason Kimberley on Dec 06 2009, Mar 19 2010, and Nov 02 2011.

A185224 Number of disconnected 2-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 15, 17, 23, 26, 33, 38, 49, 56, 69, 80, 99, 114, 139, 160, 194, 224, 268, 310, 370, 426, 504, 582, 687, 790, 927, 1066, 1247, 1433, 1667, 1913, 2222, 2545, 2944, 3369, 3888, 4442, 5112, 5833, 6697, 7631, 8739

Offset: 0

Jason Kimberley, Feb 22 2011

a(n) is also the number of partitions of n with each part at least 4 and at most n-1. The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

Andrew van den Hoeven, Table of n, a(n) for n = 0..1000
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

2-regular graphs with girth at least 4: A185114 (connected), this sequence (disconnected), A008484 (not necessarily connected).
Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).
Disconnected 2-regular simple graphs with girth at least g [partitions of n with each part i being g <= i < n]: A165652 (g=3), this sequence (g=4), A185225 (g=5), A185226 (g=6), A185227 (g=7), A185228 (g=8), A185229 (g=9).

Magma
```
A185224 := func;
```

a(n) = A008484(n) - A185114(n).

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

Original entry on oeis.org

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

A185214 Number of disconnected regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 1, 3, 2, 4, 3, 7, 5, 10, 8, 22, 12, 54, 20, 218, 62, 1436, 1731, 27810, 197981, 2613814, 33117962, 463707741, 6709514340, 102306352539, 1597440872801

Offset: 0

Jason Kimberley, Mar 26 2012

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

This sequence is the row sum sequence of A185204.
Regular graphs, of any degree, with girth at least 4: A186724 (connected), this sequence (disconnected), A185314 (not necessarily connected).
Disconnected k-regular simple graphs with girth at least 4: this sequence (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

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

A185204 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 2, 1, 1, 4, 1, 1, 0, 4, 0, 1, 1, 6, 2, 1, 0, 7, 0, 1, 1, 10, 9, 1, 1, 0, 11, 0, 0, 1, 1, 15, 35, 2, 1, 0, 17, 0, 2, 1, 1, 23, 177, 15, 1, 1, 0, 26, 0, 35, 0, 1, 1, 33, 1153, 247, 1, 1, 0, 38, 0, 1692, 0, 1, 1, 49, 10341, 17409, 8, 1

Offset: 1

Jason Kimberley, Feb 22 2011

For n >= 0 and 0 <= k <= A002265(n).

			0;
1;
1;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1, 1;
1, 0, 1;
1, 1, 2;
1, 0, 2;
1, 1, 4, 1;
1, 0, 4, 0;
1, 1, 6, 2;
1, 0, 7, 0;
1, 1, 10, 9, 1;
1, 0, 11, 0, 0;
1, 1, 15, 35, 2;
1, 0, 17, 0, 2;

Jason Kimberley, Table of i, a(i)=D(n,k) for i = 1..147 (n = 1..32)
Jason Kimberley, Incomplete table of i, n, k, D(n,k)=a(i) for n = 1..38 (i = 1..209)
Jason Kimberley, Disconnected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), this sequence (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

The b-file corrected and a-file expanded by the author, Jan 19 2013

A185234 Number of disconnected 3-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 9, 35, 177, 1153, 10341, 120523, 1702432, 27507351, 492377298, 9600913017, 201990426697, 4555849947783

Offset: 0

Jason Kimberley, Feb 22 2011

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

Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), this sequence (k=3), A185244 (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

A185254 Number of disconnected 5-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 8, 395, 407240, 1125431866, 3814677304834

Offset: 0

Jason Kimberley, Feb 22 and Nov 04 2011

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

5-regular simple graphs on 2n vertices with girth at least 4: A058275 (connected), this sequence (disconnected), A185354 (not necessarily connected).

Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), this sequence (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

A185264 Number of disconnected 6-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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 10, 7, 277, 3742, 483330, 69827771, 14836620025

Offset: 0

Jason Kimberley, Feb 22 2011

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

6-regular simple graphs with girth at least 4: A058276 (connected), this sequence (disconnected), A185364 (not necessarily connected).

Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), A185254 (k=5), this sequence (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

a(n)
= A185364(n) - A058276(n)
= Euler_transformation(A058276)(n) - A058276(n).

A185274 Number of disconnected 7-regular simple graphs on 2n 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, 1, 1, 9, 749, 2888270

Offset: 0

Jason Kimberley, Feb 22 2011

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

Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), A185254 (k=5), A185264 (k=6), this sequence (k=7), A185284 (k=8), A185294 (k=9).

A185284 Number of disconnected 8-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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 14, 1

Offset: 0

Jason Kimberley, Feb 22 2011

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

Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), this sequence (k=8), A185294 (k=9).

cp's OEIS Frontend

A033886 Number of connected 4-regular simple graphs on n vertices with girth at least 4.

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A185224 Number of disconnected 2-regular simple graphs on n vertices with girth at least 4.

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

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A185214 Number of disconnected regular simple graphs on n vertices with girth at least 4.

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A185204 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth at least 4.

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A185234 Number of disconnected 3-regular simple graphs on 2n vertices with girth at least 4.

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A185254 Number of disconnected 5-regular simple graphs on 2n vertices with girth at least 4.

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A185264 Number of disconnected 6-regular simple graphs on n vertices with girth at least 4.

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Formula

A185274 Number of disconnected 7-regular simple graphs on 2n vertices with girth at least 4.

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A185284 Number of disconnected 8-regular simple graphs on n vertices with girth at least 4.

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