A002851 -id:A002851 - cp's OEIS frontend

A014377 Number of connected regular graphs of degree 7 with 2n nodes.

1, 0, 0, 0, 1, 5, 1547, 21609301, 733351105934, 42700033549946250, 4073194598236125132578, 613969628444792223002008202, 141515621596238755266884806115631

Offset: 0

N. J. A. Sloane

			a(0)=1 because the null graph (with no vertices) is vacuously 7-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, Regular Graph
Eric Weisstein's World of Mathematics, Septic Graph

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)
7-regular simple graphs: this sequence (connected), A165877 (disconnected), A165628 (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), this sequence (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 7-regular simple graphs with girth at least g: this sequence (g=3), A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4), A184965 (g=5). (End)

a(n) = A184973(n) + A181153(n).
a(n) = A165628(n) - A165877(n).
This sequence is the inverse Euler transformation of A165628.

Added another term from Meringer's page. Dmitry Kamenetsky, Jul 28 2009
Term a(8) (on Meringer's page) was found from running Meringer's GENREG for 325 processor days at U. Newcastle by Jason Kimberley, Oct 02 2009
a(9)-a(11) from Andrew Howroyd, Mar 13 2020
a(12) from Andrew Howroyd, May 19 2020

A006923 Number of connected trivalent graphs with 2n nodes and with girth exactly 3.

Original entry on oeis.org

0, 0, 1, 1, 3, 13, 63, 399, 3268, 33496, 412943, 5883727, 94159721, 1661723296, 31954666517, 663988090257, 14814445040728

Offset: 0

N. J. A. Sloane

CRC Handbook of Combinatorial Designs, 1996, p. 647.
Gordon Royle, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g.

Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); specified g: this sequence (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).

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

a(n) = A002851(n) - A014371(n).

Definition corrected to include "connected", and "girth at least 3" minus "girth at least 4" formula provided by Jason Kimberley, Dec 12 2009

Terms a(14), a(15), and a(16) appended using "new" terms of A014371 by Jason Kimberley, Nov 16 2011

A006924 Number of connected trivalent graphs with 2n nodes and girth exactly 4.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 20, 101, 743, 7350, 91763, 1344782, 22160335, 401278984, 7885687604, 166870266608, 3781101495300

Offset: 0

N. J. A. Sloane

CRC Handbook of Combinatorial Designs, 1996, p. 647.
Gordon Royle, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Connected k-regular simple graphs with girth exactly 4: this sequence (k=3), A184944 (k=4), A184954 (k=5), A184964 (k=6), A184974 (k=7).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); specified g: A006923 (g=3), this sequence (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).
Connected 3-regular simple graphs with girth at least g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

a(n) = A014371(n) - A014372(n).

Definition corrected to include "connected", and "girth at least 4" minus "girth at least 5" formula provided by Jason Kimberley, Dec 12 2009

A014375 Number of trivalent connected simple graphs with 2n nodes and girth at least 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 21, 546, 30368, 1782840, 95079083, 4686063120, 220323447962, 10090653722861

Offset: 0

N. J. A. Sloane

nonn

The null graph on 0 vertices is vacuously connected and 3-regular; since it is acyclic, it has infinite girth. [Jason Kimberley, Jan 29 2011]

CRC Handbook of Combinatorial Designs, 1996, p. 647.

Jason Kimberley, Connected regular graphs with girth at least 7
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, May 29 2010]

From Jason Kimberley, May 29 2010 and Jan 29 2011: (Start)
Connected k-regular simple graphs with girth at least 7: A186727 (any k), A186717 (triangle); specific k: A185117 (k=2), this sequence (k=3).
Trivalent simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), this sequence (g=7), A014376 (g=8).
Trivalent 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). (End)

a(n) = A006927(n) + A014376(n).

Terms a(17), a(18), and a(19) found by running Meringer's GENREG for 1.9 hours, 99.6 hours, and 207.8 processor days, at U. Ncle., by Jason Kimberley, May 29 2010

Terms a(20) and a(21) from House of Graphs via Jason Kimberley, May 21 2017

A014376 Number of trivalent 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, 13, 155, 4337, 266362, 20807688

Offset: 0

N. J. A. Sloane

CRC Handbook of Combinatorial Designs, 1996, p. 647.

Jason Kimberley, Connected regular graphs with girth at least 8
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast Generation of Regular Graphs and Construction of Cages, Journal of Graph Theory, 30 (1999), 137-146.

Contribution from Jason Kimberley, May 18 2010 and Jan 29 2011: (Start)
Connected k-regular simple graphs with girth at least 8: A186728 (any k), A186718 (triangle); specific k: A185118 (k=2), this sequence (k=3).
Trivalent 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), this sequence (g=8).
Trivalent 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). (End)

Terms a(21), a(22), and a(23) found by running Meringer's GENREG for 0.15, 5.0, and 176.2 processor days, respectively, at U. Ncle. by Jason Kimberley, May 18 2010

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

A198303 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 1, 3, 2, 13, 5, 1, 63, 20, 2, 399, 101, 8, 1, 3268, 743, 48, 1, 33496, 7350, 450, 5, 412943, 91763, 5751, 32, 5883727, 1344782, 90553, 385, 94159721, 22160335, 1612905, 7573, 1, 1661723296, 401278984, 31297357, 181224, 3, 31954666517

Offset: 2

Jason Kimberley, Nov 16 2011

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

			1;
1, 1;
3, 2;
13, 5, 1;
63, 20, 2;
399, 101, 8, 1;
3268, 743, 48, 1;
33496, 7350, 450, 5;
412943, 91763, 5751, 32;
5883727, 1344782, 90553, 385;
94159721, 22160335, 1612905, 7573, 1;
1661723296, 401278984, 31297357, 181224, 3;
31954666517, 7885687604, 652159389, 4624480, 21;
663988090257, 166870266608, 14499780660, 122089998, 545;
14814445040728, 3781101495300, 342646718608, 3328899586, 30368;

F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A002851(n).
Connected 3-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).
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).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: this sequence (k=3), A184940 (k=4), A184950 (k=5), A184960 (k=6), A184970 (k=7), A184980 (k=8).

A000421 Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.

Original entry on oeis.org

1, 2, 6, 20, 91, 509, 3608, 31856, 340416, 4269971, 61133757, 978098997, 17228295555, 330552900516, 6853905618223, 152626436936272, 3631575281503404, 91928898608055819, 2466448432564961852, 69907637101781318907

Offset: 1

N. J. A. Sloane

a(n) is also the number of isomorphism classes of connected 3-regular simple graphs of order 2n with possibly loops. - Nico Van Cleemput, Jun 04 2014

There are no graphs of order 2n+1 satisfying the condition above. - Natan Arie Consigli, Dec 20 2019

			From _Natan Arie Consigli_, Dec 20 2019: (Start)
a(1) = 1: with two nodes the only viable option is the triple edged path multigraph.
a(2) = 4: with four nodes we have two cases: the tetrahedral graph and the square graph with single and double edges on opposite sides.
(End)

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92 [gives incorrect a(6)].
CRC Handbook of Combinatorial Designs, 1996, p. 651 [or: 2006, table 4.40].

Jan-Peter Börnsen, Anton E. M. van de Ven, Tangent Developable Orbit Space of an Octupole, arXiv:1807.04817 [hep-th], 2018.
G. Brinkmann, N. Van Cleemput, T. Pisanski, Generation of various classes of trivalent graphs, Theoretical Computer Science 502, 2013, pp.16-29.
R. J. Mathar, Cubic multigraphs A000421
Brendan McKay and others, Nauty Traces

Column k=3 of A328682 (table of k-regular n-node multigraphs).

Cf. A129416, A005967 (loops allowed), A129417, A129419, A129421, A129423, A129425, A002851 (no multiedges).

for n in {1..10}; do geng -cqD3 $[2*$n] | multig -ur3; done # Sean A. Irvine, Sep 24 2015

Inverse Euler transform of A129416. - Andrew Howroyd, Mar 19 2020

More terms from Brendan McKay, Apr 15 2007

a(13)-a(20) from Andrew Howroyd, Mar 19 2020

A006925 Number of connected trivalent graphs with 2n nodes and girth exactly 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 8, 48, 450, 5751, 90553, 1612905, 31297357, 652159389, 14499780660, 342646718608

Offset: 0

N. J. A. Sloane

CRC Handbook of Combinatorial Designs, 1996, p. 647.
Gordon Royle, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Connected k-regular simple graphs with girth exactly 5: this sequence (k=3), A184945 (k=4), A184955 (k=5).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); specified g: A006923 (g=3), A006924 (g=4), this sequence
(g=5), A006926 (g=6), A006927 (g=7).
Connected 3-regular simple graphs with girth at least g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).

a(n) = A014372(n) - A014374(n).

Definition corrected to include "connected", and "girth at least 5" minus "girth at least 6" formula provided by Jason Kimberley, Dec 12 2009

A014381 Number of connected regular graphs of degree 9 with 2n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 9, 88193, 113314233813, 281341168330848874, 1251392240942040452186674, 9854603833337765095207342173991, 134283276101750327256393048776114352985

Offset: 0

N. J. A. Sloane

Since the nontrivial 9-regular graph with the least number of vertices is K_10, there are no disconnected 9-regular graphs with less than 20 vertices. Thus for n<20 this sequence also gives the number of all 9-regular graphs on 2n vertices. - Jason Kimberley, Sep 25 2009

			The null graph on 0 vertices is vacuously connected and 9-regular; since it is acyclic, it has infinite girth. - _Jason Kimberley_, Feb 10 2011

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, Regular Graph.

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), A014378 (k=8), this sequence (k=9), A014382 (k=10), A014384 (k=11).
9-regular simple graphs: this sequence (connected), A185293 (disconnected).
Connected 9-regular simple graphs with girth at least g: this sequence (g=3), A181170 (g=4).
Connected 9-regular simple graphs with girth exactly g: A184993 (g=3).

a(n) = A184993(n) + A181170(n).

a(8) appended using the symmetry of A051031 by Jason Kimberley, Sep 25 2009
a(9)-a(10) from Andrew Howroyd, Mar 13 2020
a(10) corrected and a(11)-a(12) from Andrew Howroyd, May 19 2020

cp's OEIS Frontend

A014377 Number of connected regular graphs of degree 7 with 2n nodes.

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A006923 Number of connected trivalent graphs with 2n nodes and with girth exactly 3.

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A006924 Number of connected trivalent graphs with 2n nodes and girth exactly 4.

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A014375 Number of trivalent connected simple graphs with 2n nodes and girth at least 7.

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A014376 Number of trivalent connected simple graphs with 2n nodes and girth at least 8.

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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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A198303 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth exactly g.

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A000421 Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.

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nauty

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A006925 Number of connected trivalent graphs with 2n nodes and girth exactly 5.

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