A006823 -id:A006823 - cp's OEIS frontend

A008326 Triangle read by rows: T(n,k) is the number of simple regular connected bipartite graphs with 2n nodes and degree k, (2 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 13, 14, 4, 1, 1, 1, 38, 129, 41, 7, 1, 1, 1, 149, 1980, 1981, 157, 8, 1, 1, 1, 703, 62611, 304495, 62616, 725, 12, 1, 1, 1, 4132, 2806490, 78322915, 78322916, 2806508, 4196, 14, 1, 1, 1, 29579, 158937213, 27033154060, 147252447227, 27033154065, 158937367, 29817, 21, 1, 1

Offset: 2

Brendan McKay and Eric Rogoyski

This sequence can be derived from A133687 and A333159. In particular, if w(n) is the inverse Euler transform of column k of A133687 and s(n) is the inverse Euler transform of column k of A333159, then 2*T(2*n+1,k) = w(2*n+1) + s(2*n+1) and 2*T(2*n,k) = w(2*n) + s(2*n) - w(n) + T(n,k). - Andrew Howroyd, Apr 03 2020

			Triangle begins:
  1;
  1,   1;
  1,   1,    1;
  1,   2,    1,    1;
  1,   5,    4,    1,   1;
  1,  13,   14,    4,   1, 1;
  1,  38,  129,   41,   7, 1, 1;
  1, 149, 1980, 1981, 157, 8, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..154
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Columns k=3..7 are A006823, A006824, A006825, A014385, A014387.
Row sums are in A008323.
Cf. A008327, A133687, A333159.

More terms from Eric Rogoyski, May 15 1997
Name clarified by Andrew Howroyd, Sep 05 2018
Terms a(55) and beyond from Andrew Howroyd, Apr 03 2020

A008325 Number of simple regular trivalent bipartite graphs with 2n nodes.

Original entry on oeis.org

1, 1, 2, 6, 14, 41, 157, 725, 4196, 29816, 246644, 2297075, 23503477, 260265023, 3090336300, 39101547971, 524782945991, 7443247863498, 111221983956652, 1746165682538497, 28734206614035245, 494526496354065244, 8883865784392246280, 166286434745252091055, 3237719048384605059117, 65477287940472122129194

Offset: 3

Brendan McKay

Euler transform of A006823. - Peter J. Taylor, Sep 28 2017

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Sean A. Irvine, On the difference between A004066 and A008325

Column k=3 of A008327.

Cf. A004066 (bicolored), A006823 (connected).

A006823 = Cases[Import["https://oeis.org/A006823/b006823.txt", "Table"], {, }][[All, 2]];
etr[f_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d f[d], {d, Divisors[j]}] b[n - j], {j, 1, n}]/n]; b];
b[n_] := If[n >= 3, A006823[[n - 2]], 0];
a = etr[b];
a /@ Range[3, 16] (* Jean-François Alcover, Dec 03 2019 *)

a(15)-a(16) from Peter J. Taylor, Sep 28 2017

Terms a(17) and beyond from Andrew Howroyd, Apr 03 2020

A004066 Number of simple regular trivalent bicolored graphs with 2n nodes.

Original entry on oeis.org

0, 0, 1, 1, 2, 6, 14, 41, 157, 725, 4196, 29817, 246646, 2297088, 23503564, 260265650, 3090341095, 39101587595, 524783295041, 7443251159470, 111222017297268, 1746166043555813, 28734210790531045, 494526547845483641, 8883866458982018870, 166286444108288113541, 3237719185652343485853, 65477290060076644381373

Offset: 1

Gunnar Brinkmann, Brendan McKay and Eric Rogoyski

nonn

Sean A. Irvine, On the difference between A004066 and A008325
Patric R. J. Östergård, Classifying generalized Howell designs, Designs Codes Cryptog. (2025). See Sect. 3.2.1.

Cf. A000512, A000840, A008325 (bipartite), A006823 (connected).

a(n) = (A000840(n) + A000512(n))/2. - Andrew Howroyd, Apr 01 2020

a(1)-a(2) prepended and terms a(15) and beyond from Andrew Howroyd, Apr 01 2020

A006824 Number of connected regular bipartite graphs of degree 4 with 2n nodes.

Original entry on oeis.org

1, 1, 4, 14, 129, 1980, 62611, 2806490, 158937213, 10773251972, 855658082615, 78558949838723, 8251166737356319, 982806379842257309, 131756174189661102281, 19748565896506014747623, 3289970433888731383271400, 605948436052375098046655323, 122796503871896458570959144266

Offset: 4

N. J. A. Sloane

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.
H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
M. Meringer, Tables of Regular Graphs

Column 4 of A008326.

Cf. A006823, A006825, A333730.

Inverse Euler transform of A333730. - Andrew Howroyd, Apr 03 2020

Terms a(12) and beyond from Andrew Howroyd, Apr 03 2020

A006825 Number of connected regular bipartite graphs of degree 5 with 2n nodes.

Original entry on oeis.org

1, 1, 4, 41, 1981, 304495, 78322915, 27033154060, 11934777413004, 6593485023087880, 4485517185017428244, 3707462300996566329965, 3680029088808677189795063, 4341813441626419937873317841, 6033239205199247162872884404386, 9792722283774374706204648839780148

Offset: 5

N. J. A. Sloane

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Meringer, Tables of Regular Graphs
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Column 5 of A008326.

Cf. A006823, A006824, A333731.

Inverse Euler transform of A333731. - Andrew Howroyd, Apr 03 2020

More terms from Eric Rogoyski, May 15 1997

Terms a(12) and beyond from Andrew Howroyd, Apr 03 2020

A260811 Number of trivalent bipartite connected simple graphs with 2n nodes and girth at least 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 10, 28, 162, 1201, 11415, 125571, 1514489

Offset: 0

Dylan Thurston, Jul 31 2015

The null graph on 0 vertices is vacuously connected, 3-regular, and bipartite; since it is acyclic, it has infinite girth.

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
House of Graphs, Cubic bipartite graphs

Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

Connected bipartite trivalent simple graphs with girth at least g: A006823 (g=4), this sequence (g=6), A260813 (g=8).

A260813 Number of trivalent bipartite connected simple graphs with 2n nodes and girth at least 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 10, 101, 2510, 79605, 2607595, 81716416, 2472710752

Offset: 0

Dylan Thurston, Jul 31 2015

The null graph on 0 vertices is vacuously connected, 3-regular, and bipartite; since it is acyclic, it has infinite girth.

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
G. Brinkmann, J. Goedgebeur and B.D. McKay, The Minimality of the Georges-Kelmans Graph, arXiv:2101.00943 [math.CO], 2021.
House of Graphs, Cubic bipartite graphs

Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

Connected bipartite trivalent simple graphs with girth at least g: A006823 (g=4), A260811 (g=6), this sequence (g=8).

a(23)-a(24) from the House-of-Graphs added by R. J. Mathar, Sep 29 2017

a(25)-a(26) from Jan Goedgebeur, Aug 17 2021

A358282 Number of connected bipartite cubic graphs with 2*n nodes and exactly one edge-Kempe equivalence class.

Original entry on oeis.org

0, 1, 0, 2, 1, 6, 4, 24, 28, 140, 244, 1026

Offset: 3

N. J. A. Sloane, Nov 08 2022

The reference gives further terms.

Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 2.

Cf. A006823.

A358283 Number of connected bipartite cubic graphs with 2*n nodes and the maximum number of edge-Kempe equivalence classes.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 7, 13, 25, 67, 111, 453, 588, 3112, 3469, 22832

Offset: 3

N. J. A. Sloane, Nov 08 2022

Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 2.

Cf. A006823.

cp's OEIS Frontend

A008326 Triangle read by rows: T(n,k) is the number of simple regular connected bipartite graphs with 2n nodes and degree k, (2 <= k <= n).

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A008325 Number of simple regular trivalent bipartite graphs with 2n nodes.

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A004066 Number of simple regular trivalent bicolored graphs with 2n nodes.

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A006824 Number of connected regular bipartite graphs of degree 4 with 2n nodes.

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A006825 Number of connected regular bipartite graphs of degree 5 with 2n nodes.

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A260811 Number of trivalent bipartite connected simple graphs with 2n nodes and girth at least 6.

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A260813 Number of trivalent bipartite connected simple graphs with 2n nodes and girth at least 8.

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A358282 Number of connected bipartite cubic graphs with 2*n nodes and exactly one edge-Kempe equivalence class.

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A358283 Number of connected bipartite cubic graphs with 2*n nodes and the maximum number of edge-Kempe equivalence classes.

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Author

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