A184981 -id:A184981 - 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

A185131 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 2, 1, 5, 2, 19, 6, 1, 85, 22, 2, 509, 110, 9, 1, 4060, 792, 49, 1, 41301, 7805, 455, 5, 510489, 97546, 5783, 32, 7319447, 1435720, 90938, 385, 117940535, 23780814, 1620479, 7574, 1, 2094480864, 432757568, 31478584, 181227, 3, 40497138011, 8542471494

Offset: 2

Jason Kimberley, Jan 09 2012

The first column is for girth at least 3. The row length is incremented to g-2 when 2n reaches A000066(g).

			                  1;
                  2,             1;
                  5,             2;
                 19,             6,            1;
                 85,            22,            2;
                509,           110,            9,          1;
               4060,           792,           49,          1;
              41301,          7805,          455,          5;
             510489,         97546,         5783,         32;
            7319447,       1435720,        90938,        385;
          117940535,      23780814,      1620479,       7574,         1;
         2094480864,     432757568,     31478584,     181227,         3;
        40497138011,    8542471494,    656783890,    4624501,        21;
       845480228069,  181492137812,  14621871204,  122090544,       546,    1;
     18941522184590, 4127077143862, 345975648562, 3328929954,     30368,    0;
    453090162062723,        ?,            ?,     93990692595,   1782840,    1;
  11523392072541432,        ?,            ?,   2754222605376,  95079083,    3;
 310467244165539782,        ?,            ?,          ?,     4686063120,   13;
8832736318937756165,        ?,            ?,          ?,   220323447962,  155;
          ?,                ?,            ?,          ?, 10090653722861, 4337;

Jason Kimberley, Table of i, a(i) for i = 2..59 (n = 2..16)
B. Brinkmann, J. Goedgebeur, and B. D. McKay, Generation of cubic graphs, Discr. Math. Theor. Comp. Sci. 13 (2) (2011) 69-80.
House of Graphs, Cubic graphs
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs

Connected 3-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: this sequence (k=3), A184941 (k=4), A184951 (k=5), A184961 (k=6), A184971 (k=7), A184981 (k=8).

Terms C(18,6), C(20,7) and C(21,7) from House of Graphs via Jason Kimberley, May 21 2017

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).

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).

A184941 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth at least g.

Original entry on oeis.org

1, 1, 2, 6, 1, 16, 0, 59, 2, 265, 2, 1544, 12, 10778, 31, 88168, 220, 805491, 1606, 8037418, 16828, 86221634, 193900, 985870522, 2452818, 11946487647, 32670330, 1, 152808063181, 456028474, 2, 2056692014474, 6636066099, 8, 28566273166527, 100135577747, 131

Offset: 5

Jason Kimberley, Jan 10 2012

The first column is for girth at least 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4. The row length is incremented to g-2 when n reaches A037233(g).

			1;
1;
2;
6, 1;
16, 0;
59, 2;
265, 2;
1544, 12;
10778, 31;
88168, 220;
805491, 1606;
8037418, 16828;
86221634, 193900;
985870522, 2452818;
11946487647, 32670330, 1;
152808063181, 456028474, 2;
2056692014474, 6636066099, 8;
28566273166527, 100135577747, 131;

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

Connected 4-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184940 (triangle); chosen g: A184943 (g=3), A184944 (g=4), A184945 (g=5), A184946 (g=6).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: A185131 (k=3), this sequence (k=4), A184951 (k=5), A184961 (k=6), A184971 (k=7), A184981 (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

A184961 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth at least g.

Original entry on oeis.org

1, 1, 4, 21, 266, 7849, 1, 367860, 0, 21609300, 1, 1470293675, 1, 113314233808, 9, 9799685588936, 6

Offset: 7

Jason Kimberley, Jan 10 2012

The first column is for girth at least 3. The row length sequence starts: 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3. The row length is incremented to g-2 when n reaches A054760(6,g).

			Triangle begins:
1;
1;
4;
21;
266;
7849, 1;
367860, 0;
21609300, 1;
1470293675, 1;
113314233808, 9;
9799685588936, 6;

Andries E. Brouwer, Cages
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g

Connected 6-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A006822 (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: A184960 (triangle); chosen g: A184963 (g=3), A184964 (g=4).
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), this sequence (k=6), A184971 (k=7), A184981 (k=8).

A184971 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 5, 1547, 21609301, 1, 733351105934, 1

Offset: 4

Jason Kimberley, Jan 10 2012

The first column is for girth at least 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2. The row length is incremented to g-2 when 2n reaches A054760(7,g).

			1;
5;
1547;
21609301, 1;
733351105934, 1;
?, 8;
?, 741;
?, 2887493;

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

Connected 7-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A014377 (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: A184970 (triangle); chosen g: A184973 (g=3), A184974 (g=4).
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), this sequence (k=7), A184981 (k=8).

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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A185131 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth at least g.

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

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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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A184941 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth at least 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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A184961 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth at least g.

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A184971 Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth at least g.

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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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