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A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.
(Formerly M1617)
1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, 265, 1544, 10778, 88168, 805491, 8037418, 86221634, 985870522, 11946487647, 152808063181, 2056692014474, 29051272833609, 429668180677439, 6640165204855036, 107026584471569605, 1796101588825595008, 31333997930603283531, 567437240683788292989 (list; graph; refs; listen; history; text; internal format)



The null graph on 0 vertices is vacuously connected and 4-regular. - Jason Kimberley, Jan 29 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.

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..28.

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, Nov 24 2009]

M. Meringer, GenReg, Generation of regular graphs, program.

Markus Meringer, H. James Cleaves, Stephen J. Freeland, Beyond Terrestrial Biology: Charting the Chemical Universe of α-Amino Acid Structures, Journal of Chemical Information and Modeling, 53.11 (2013), pp. 2851-2862.

Eric Weisstein's World of Mathematics, Connected Graph

Eric Weisstein's World of Mathematics, Quartic Graph

Eric Weisstein's World of Mathematics, Regular Graph

Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.


a(n) = A184943(n) + A033886(n).

a(n) = A033301(n) - A033483(n).

Inverse Euler transform of A033301.


From Jason Kimberley, Mar 27 2010 and Jan 29 2011: (Start)

4-regular simple graphs: this sequence (connected), A033483 (disconnected), A033301 (not necessarily connected).

Connected regular simple graphs: A005177 (any degree), A068934 (triangular array); specified degree k: A002851 (k=3), this sequence (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).

Connected 4-regular simple graphs with girth at least g: this sequence (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).

Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5).

Connected 4-regular graphs: this sequence (simple), A085549 (multigraphs with loops allowed), A129417  (multigraphs with loops verboten). (End)

Sequence in context: A221841 A068787 A073959 * A131385 A027742 A324062

Adjacent sequences:  A006817 A006818 A006819 * A006821 A006822 A006823




N. J. A. Sloane


a(19)-a(22) were appended by Jason Kimberley on Sep 04 2009, Nov 24 2009, Mar 27 2010, and Mar 18 2011, from running M. Meringer's GENREG for 3.4, 44, and 403 processor days, and 15.5 processor years, at U. Ncle.

a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 10 2020



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Last modified April 13 10:24 EDT 2021. Contains 342935 sequences. (Running on oeis4.)