A014371 Number of trivalent connected simple graphs with 2n nodes and girth at least 4.

1, 0, 0, 1, 2, 6, 22, 110, 792, 7805, 97546, 1435720, 23780814, 432757568, 8542471494, 181492137812, 4127077143862

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more
• hard

Comments

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]

References

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

Links

• G. Brinkmann, J. Goedgebeur and B. D. McKay, Generation of Cubic graphs, Discrete Mathematics and Theoretical Computer Science, 13 (2) (2011), 69-80. (hal-00990486)
• House of Graphs, Cubic graphs.
• 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.
• M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.

Crossrefs

Contribution from Jason Kimberley, Jun 28 2010 and Jan 29 2011: (Start)

3-regular simple graphs with girth at least 4: this sequence (connected), A185234 (disconnected), A185334 (not necessarily connected).

Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), this sequence (k=3), A033886 (k=4), A058275 (k=5), A058276 (k=6), A181153 (k=7), A181154 (k=8), A181170 (k=9).

Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), this sequence (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). (End)

Programs

Mathematica

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A002851 = A@002851;
A006923 = A@006923;
a[n_] := A002851[[n + 1]] - A006923[[n + 1]];
a /@ Range[0, 16] (* Jean-François Alcover, Jan 27 2020 *)

Extensions

Terms a(14) and a(15) appended, from running Meringer's GENREG for 4.2 and 93.2 processor days at U. Newcastle, by Jason Kimberley on Jun 28 2010.

a(16), from House of Graphs, by Jan Goedgebeur et al., added by Jason Kimberley, Feb 15 2011