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A006821 Number of connected regular graphs of degree 5 (or quintic graphs) with 2n nodes.
(Formerly M3168)
1, 0, 0, 1, 3, 60, 7848, 3459383, 2585136675, 2807105250897, 4221456117363365, 8516994770090547979, 22470883218081146186209, 75883288444204588922998674, 322040154704144697047052726990 (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.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

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..14.

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

Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.

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, Nov 24 2009]

Eric Weisstein's World of Mathematics, Quintic Graph

Eric Weisstein's World of Mathematics, Regular Graph


a(n) = A184953(n) + A058275(n).

a(n) = A165626(n) - A165655(n).

Inverse Euler transform of A165626.


a(0)=1 because the null graph (with no vertices) is vacuously 5-regular and connected.


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

5-regular simple graphs: this sequence (connected), A165655 (disconnected), A165626 (not necessarily connected).

Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), this sequence (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).

Connected 5-regular simple graphs with girth at least g: this sequence (g=3), A058275 (g=4), A205295 (g=5).

Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), A184954 (g=4), A184955 (g=5).

Connected 5-regular graphs: A129430 (loops and multiple edges allowed), A129419 (no loops but multiple edges allowed), this sequence (no loops nor multiple edges). (End)

Sequence in context: A202065 A036770 A201699 * A165626 A120307 A022915

Adjacent sequences:  A006818 A006819 A006820 * A006822 A006823 A006824




N. J. A. Sloane


By running M. Meringer's GENREG for about 2 processor years at U. Newcastle, a(9) was found by Jason Kimberley, Nov 24 2009

a(10)-a(14) from Andrew Howroyd, Mar 10 2020



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Last modified April 16 04:49 EDT 2021. Contains 343030 sequences. (Running on oeis4.)