A184983 -id:A184983 - cp's OEIS frontend

A014378 Number of connected regular graphs of degree 8 with n nodes.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848873, 214755319657939505395, 187549729101764460261498, 186685399408147545744203815, 210977245260028917322933154987

Offset: 0

N. J. A. Sloane

Since the nontrivial 8-regular graph with the least number of vertices is K_9, there are no disconnected 8-regular graphs with less than 18 vertices. Thus for n<18 this sequence is identical to A180260. - Jason Kimberley, Sep 25 2009 and Feb 10 2011

			a(0)=1 because the null graph (with no vertices) is vacuously 8-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.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Octic Graph
Eric Weisstein's World of Mathematics, Regular Graph

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
8-regular simple graphs: this sequence (connected), A165878 (disconnected), A180260 (not necessarily connected).
Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), this sequence (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3). (End)

a(n) = A184983(n) + A181154(n).
a(n) = A180260(n) + A165878(n).
This sequence is the inverse Euler transformation of A180260.

Using the symmetry of A051031, a(15) and a(16) were appended by Jason Kimberley, Sep 25 2009

a(17)-a(22) from Andrew Howroyd, Mar 13 2020

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

A181154 Number of connected 8-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 13, 1

Offset: 0

Jason Kimberley, week to Jan 31 2011

a(20) and a(21) were computed by the author, using GENREG, over 79 processor hours and 294 processor days, respectively, during Dec 2009.

			The a( 0)=1 null graph is vacuously 8-regular and connected; since it is acyclic then it has infinite girth.
The a(16)=1 graph is the complete bipartite graph K_{8,8}.
The a(21)=1 graph has girth 4, automorphism group of order 829440, and the following adjacency lists:
01 : 02 03 04 05 06 07 08 09
02 : 01 10 11 12 13 14 15 16
03 : 01 10 11 12 13 14 15 16
04 : 01 10 11 12 13 14 15 16
05 : 01 10 11 12 13 14 15 16
06 : 01 10 11 12 17 18 19 20
07 : 01 10 11 13 17 18 19 20
08 : 01 10 12 13 17 18 19 20
09 : 01 11 12 13 17 18 19 20
10 : 02 03 04 05 06 07 08 21
11 : 02 03 04 05 06 07 09 21
12 : 02 03 04 05 06 08 09 21
13 : 02 03 04 05 07 08 09 21
14 : 02 03 04 05 17 18 19 20
15 : 02 03 04 05 17 18 19 20
16 : 02 03 04 05 17 18 19 20
17 : 06 07 08 09 14 15 16 21
18 : 06 07 08 09 14 15 16 21
19 : 06 07 08 09 14 15 16 21
20 : 06 07 08 09 14 15 16 21
21 : 10 11 12 13 17 18 19 20

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

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

8-regular simple graphs with girth at least 4: this sequence (connected), A185284 (disconnected), A185384 (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), A058275 (k=5), A058276 (k=6), A181153 (k=7), this sequence (k=8), A181170 (k=9).
Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), this sequence (g=4).
Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3).

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

Jason Kimberley, Feb 27 2011

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g
Eric Weisstein's World of Mathematics, Girth.
Eric Weisstein's World of Mathematics, Quintic Graph.

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).

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

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.

A184981 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth at least g.

Original entry on oeis.org

1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 1

Offset: 9

Jason Kimberley, Jan 19 2012

The first column is for girth at least 3. The row length is incremented to g-2 when n reaches A054760(8,g).

			1;
1;
6;
94;
10786;
3459386;
1470293676;
733351105935, 1;
?, 0;
?, 1;
?, 0;
?, 13;
?, 1;

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

Connected 8-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3).
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), A184971 (k=7), this sequence (k=8), A184991 (k=9).

A184980 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105934, 1

Offset: 9

Jason Kimberley, Jan 19 2012

The first column is for girth at least 3. The row length is incremented to g-2 when 2n reaches A054760(8,g).

			1;
1;
6;
94;
10786;
3459386;
1470293676;
733351105934, 1;
?, 0;
?, 1;
?, 0;
?, 13;
?, 1;

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

Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184983 (g=3).
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), A184970 (k=7), this sequence (k=8).

A186743 Number of connected regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 13, 21, 157, 536, 18942, 389404, 50314456, 2942196832, 1698517018391

Offset: 0

Jason Kimberley, Dec 01 2011

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

Connected k-regular simple graphs with girth exactly 3: this sequence (any k), A186733 (triangular array); specified k: A006923 (k=3),A184943 (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7),A184983 (k=8), A184993 (k=9).

a(n) = A005177(n) - A186724(n).

A184991 Irregular triangle C(n,g) counting the connected 9-regular simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 9, 88193, 113314233813

Offset: 5

Jason Kimberley, Feb 03 2012

The first column is for girth at least 3. The row length is incremented to g-2 when 2n reaches A054760(9,g).

			1;
 9;
 88193;
 113314233813;
 ?, 1;
 ?, 1;
 ?, 14;

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

Connected 9-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A014381 (g=3), A181170 (g=4).
Connected 9-regular simple graphs with girth exactly g: A184990 (triangle); chosen g: A184983 (g=3).
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), A184971 (k=7), A184981 (k=8), this sequence (k=9).

cp's OEIS Frontend

A014378 Number of connected regular graphs of degree 8 with n nodes.

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

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A181154 Number of connected 8-regular simple graphs on n vertices with girth at least 4.

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A184953 Number of connected 5-regular (or quintic) simple graphs on 2n 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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A184981 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth at least g.

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A184980 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth exactly g.

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A186743 Number of connected regular simple graphs on n vertices with girth exactly 3.

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A184991 Irregular triangle C(n,g) counting the connected 9-regular simple graphs on 2n vertices with girth at least g.

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