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A006822 Number of connected regular graphs of degree 6 with n nodes.
(Formerly M3579)
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 (list; graph; refs; listen; history; text; internal format)



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).


Table of n, a(n) for n=0..24.

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


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

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

This sequence is the inverse Euler transformation of A165627.


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)

Sequence in context: A184963 A185163 A184961 * A165627 A324954 A198050

Adjacent sequences:  A006819 A006820 A006821 * A006823 A006824 A006825




N. J. A. Sloane


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



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