A006822 Number of connected regular graphs of degree 6 (or sextic graphs) with n nodes.

1, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7849, 367860, 21609300, 1470293675, 113314233808, 9799685588936, 945095823831036, 101114579937187980, 11945375659139626688, 1551593789610509806552, 220716215902792573134799, 34259321384370620122314325, 5782740798229825207562109439

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

References

• 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).

Links

• 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, Regular Graph
• Eric Weisstein's World of Mathematics, Sextic Graph

Crossrefs

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)

6-regular simple graphs: this sequence (connected), A165656 (disconnected), A165627 (not necessarily connected).

Connected regular graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), this sequence (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).

Connected 6-regular simple graphs with girth at least g: this sequence (g=3), A058276 (g=4).

Connected 6-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4). (End)

Formula

a(n) = A184963(n) + A058276(n).

a(n) = A165627(n) - A165656(n).

This sequence is the inverse Euler transformation of A165627.

Extensions

a(16) and a(17) appended, from running M. Meringer's GENREG at U. Newcastle for 41 processor days and 3.5 processor years, by Jason Kimberley, Sep 04 2009 and Nov 13 2009.

Terms a(18)-a(24), due to the extension of A165627 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020