A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

Comments

The null graph on 0 vertices is vacuously connected and 4-regular. - Jason Kimberley, Jan 29 2011

The Multiset Transform of this sequence gives a triangle which gives in row n and column k the 4-regular simple graphs with n>=1 nodes and k>=1 components (row sums A033301), starting:

;

;

;

;

1 ;

1 ;

2 ;

6 ;

16 ;

59 1 ;

265 1 ;

1544 3 ;

10778 8 ;

88168 25 ;

805491 87 1 ;

8037418 377 1 ;

86221634 2023 3 ;

985870522 13342 9 ;

11946487647 104568 27 ;

152808063181 930489 96 1 ; - R. J. Mathar, Jun 02 2022

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.
• 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).

Links

• Wayne Barrett, Shaun Fallat, Veronika Furst, Shahla Nasserasr, Brendan Rooney, and Michael Tait, Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues, arXiv:2305.10562 [math.CO], 2023. See p. 7.
• 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, and Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.

Crossrefs

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)

Formula

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

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

Inverse Euler transform of A033301.

Row sums of A184940. - R. J. Mathar, May 30 2022

Extensions

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