A184953 -id:A184953 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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).

Leonard Chidiebere Eze, Robert Jajcay, and Jorik Jooken, On (k,g)-Graphs without (g+1)-Cycles, arXiv:2411.19023 [math.CO], 2024. See p. 18.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
Markus Meringer, Tables of Regular Graphs
Markus Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
Eric Weisstein's World of Mathematics, Quintic Graph
Eric Weisstein's World of Mathematics, Regular Graph

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)

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

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

N. J. A. Sloane, Dec 17 2000

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

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

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)

a(n) = A185354(n) - A185254(n);

This sequence is the inverse Euler transformation of A185354. - Jason Kimberley, Nov 04 2011

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

A184943 Number of connected 4-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 5, 16, 57, 263, 1532, 10747, 87948, 803885, 8020590, 86027734, 983417704, 11913817317, 152352034707, 2050055948375, 28951137255862, 428085461764471

Offset: 0

Jason Kimberley, Jan 25 2011

			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(5)=1 complete graph on 5 vertices is 4-regular; it has 10 edges and 10 triangles.

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

4-regular simple graphs with girth exactly 3: this sequence (connected), A185043 (disconnected), A185143 (not necessarily connected).
Connected k-regular simple graphs with girth exactly 3: A006923 (k=3), this sequence (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).
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: this sequence (g=3), A184944 (g=4), A184945 (g=5).

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A006820 = A@006820; A033886 = A@033886;
a[n_] := A006820[[n + 1]] - A033886[[n + 1]];
a /@ Range[0, 22] (* Jean-François Alcover, Jan 27 2020 *)

a(n) = A006820(n) - A033886(n).

Term a(22) corrected and a(23) appended, due to the correction and extension of A006820 by Andrew Howroyd, from Jason Kimberley, Mar 13 2020

A186733 Triangular array C(n,r) = number of connected r-regular graphs, having girth exactly 3, with n nodes, for 0 <= r < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 5, 3, 1, 1, 0, 0, 0, 0, 16, 0, 4, 0, 1, 0, 0, 0, 13, 57, 59, 21, 5, 1, 1, 0, 0, 0, 0, 263, 0, 266, 0, 6, 0, 1, 0, 0, 0, 63, 1532, 7847, 7848, 1547, 94, 9, 1, 1, 0, 0, 0, 0, 10747, 0, 367860, 0, 10786

Offset: 1

Jason Kimberley, Mar 26 2012

			01: 0 ;
02: 0, 0 ;
03: 0, 0, 1 ;
04: 0, 0, 0, 1 ;
05: 0, 0, 0, 0, 1 ;
06: 0, 0, 0, 1, 1, 1 ;
07: 0, 0, 0, 0, 2, 0, 1 ;
08: 0, 0, 0, 3, 5, 3, 1, 1 ;
09: 0, 0, 0, 0, 16, 0, 4, 0, 1 ;
10: 0, 0, 0, 13, 57, 59, 21, 5, 1, 1 ;
11: 0, 0, 0, 0, 263, 0, 266, 0, 6, 0, 1 ;
12: 0, 0, 0, 63, 1532, 7847, 7848, 1547, 94, 9, 1, 1 ;
13: 0, 0, 0, 0, 10747, 0, 367860, 0, 10786, 0, 10, 0, 1 ;
14: 0, 0, 0, 399, 87948, 3459376, 21609299, 21609300, 3459386, 88193, 540, 13, 1, 1 ;
15: 0, 0, 0, 0, 803885, 0, 1470293674, 0, 1470293676, 0, 805579, 0, 17, 0, 1 ;
16: 0, 0, 0, 3268, 8020590, 2585136287, 113314233799, 733351105933, 733351105934, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Jason Kimberley, Table of i, a(i)=C(n,r) for i = 1..136 (n = 1..16)
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

The sum of the n-th row is A186743(n).
Connected k-regular simple graphs with girth exactly 3: this sequence (triangle), A186743 (any k); chosen k: A006923 (k=3), A184943 (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *at least* g: A068934 (g=3), A186714 (g=4), A186715 (g=5), A186716 (g=6), A186717 (g=7), A186718 (g=8), A186719 (g=9).
Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: this sequence  (g=3), A186734 (g=4).

C(n,r) = A068934(n,r) - A186714(n,r), noting that A186714 has 0 <= r <= n div 2.

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

Jason Kimberley, Feb 27 2011

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

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).

A184955 Number of connected 5-regular simple graphs on 2n 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, 4, 90

Offset: 0

Jason Kimberley, Feb 27 2011

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

Connected k-regular simple graphs with girth exactly 5: A185015 (k=2), A006925 (k=3), A184945 (k=4), this sequence (k=5).
Connected 5-regular simple graphs with girth at least g: A006821 (g=3), A058275 (g=4), A205295 (g=5).
Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), A184954 (g=4), this sequence (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

Jason Kimberley, Jan 25 2012

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

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

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

Jason Kimberley, Feb 24 2011

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).

			1;
3;
59, 1;
7847, 1;
3459376, 7;
2585136287, 388;
2807104844073, 406824;
?, 1125022325;
?, 3813549359274;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g
Jason Kimberley, Incomplete table of i, n, g, C(n,g)=a(i) for row n = 3..22

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

Jason Kimberley, Jan 10 2012

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).

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

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

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).

a(14) from Jason Kimberley, Dec 26 2012

A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 3, 59, 7848, 3459379, 2585136353, 2807104852102

Offset: 0

Jason Kimberley, Mar 12 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), this sequence (k=5), A185163 (k=6).

a(n) = A165626(n) - A185354(n).

a(n) = A184953(n) + A185053(n).

cp's OEIS Frontend

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

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

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A184943 Number of connected 4-regular simple graphs on n vertices with girth exactly 3.

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A186733 Triangular array C(n,r) = number of connected r-regular graphs, having girth exactly 3, with n nodes, for 0 <= r < n.

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

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A184955 Number of connected 5-regular simple graphs on 2n vertices with girth exactly 5.

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

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

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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.

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A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

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