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%I A006820 M1617 #90 May 24 2023 13:14:49
%S A006820 1,0,0,0,0,1,1,2,6,16,59,265,1544,10778,88168,805491,8037418,86221634,
%T A006820 985870522,11946487647,152808063181,2056692014474,29051272833609,
%U A006820 429668180677439,6640165204855036,107026584471569605,1796101588825595008,31333997930603283531,567437240683788292989
%N A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.
%C A006820 The null graph on 0 vertices is vacuously connected and 4-regular. - _Jason Kimberley_, Jan 29 2011
%C A006820 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:
%C A006820 ;
%C A006820 ;
%C A006820 ;
%C A006820 ;
%C A006820 1 ;
%C A006820 1 ;
%C A006820 2 ;
%C A006820 6 ;
%C A006820 16 ;
%C A006820 59 1 ;
%C A006820 265 1 ;
%C A006820 1544 3 ;
%C A006820 10778 8 ;
%C A006820 88168 25 ;
%C A006820 805491 87 1 ;
%C A006820 8037418 377 1 ;
%C A006820 86221634 2023 3 ;
%C A006820 985870522 13342 9 ;
%C A006820 11946487647 104568 27 ;
%C A006820 152808063181 930489 96 1 ;  - _R. J. Mathar_, Jun 02 2022
%D A006820 CRC Handbook of Combinatorial Designs, 1996, p. 648.
%D A006820 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.
%D A006820 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A006820 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006820 Wayne Barrett, Shaun Fallat, Veronika Furst, Shahla Nasserasr, Brendan Rooney, and Michael Tait, <a href="https://arxiv.org/abs/2305.10562">Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues</a>, arXiv:2305.10562 [math.CO], 2023. See p. 7.
%H A006820 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_k-reg_girth_ge_g_index">Index of sequences counting connected k-regular simple graphs with girth at least g</a>
%H A006820 M. Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html">Tables of Regular Graphs</a>
%H A006820 M. Meringer, <a href="http://dx.doi.org/10.1002/(SICI)1097-0118(199902)30:2&lt;137::AID-JGT7&gt;3.0.CO;2-G">Fast generation of regular graphs and construction of cages</a>, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
%H A006820 M. Meringer, <a href="https://sourceforge.net/projects/genreg/">GenReg</a>, Generation of regular graphs, program.
%H A006820 Markus Meringer, H. James Cleaves, Stephen J. Freeland, <a href="https://doi.org/10.1021/ci400209n">Beyond Terrestrial Biology: Charting the Chemical Universe of α-Amino Acid Structures</a>, Journal of Chemical Information and Modeling, 53.11 (2013), pp. 2851-2862.
%H A006820 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>
%H A006820 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuarticGraph.html">Quartic Graph</a>
%H A006820 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RegularGraph.html">Regular Graph</a>
%H A006820 Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, and Yuefan Deng, <a href="https://arxiv.org/abs/1907.12455">A new record of enumeration of regular graphs by parallel processing</a>, arXiv:1907.12455 [cs.DM], 2019.
%F A006820 a(n) = A184943(n) + A033886(n).
%F A006820 a(n) = A033301(n) - A033483(n).
%F A006820 Inverse Euler transform of A033301.
%F A006820 Row sums of A184940. - _R. J. Mathar_, May 30 2022
%Y A006820 From _Jason Kimberley_, Mar 27 2010 and Jan 29 2011: (Start)
%Y A006820 4-regular simple graphs: this sequence (connected), A033483 (disconnected), A033301 (not necessarily connected).
%Y A006820 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).
%Y A006820 Connected 4-regular simple graphs with girth at least g: this sequence (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
%Y A006820 Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5).
%Y A006820 Connected 4-regular graphs: this sequence (simple), A085549 (multigraphs with loops allowed), A129417  (multigraphs with loops verboten). (End)
%K A006820 nonn,nice,hard
%O A006820 0,8
%A A006820 _N. J. A. Sloane_
%E A006820 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.
%E A006820 a(22) corrected and a(23)-a(28) from _Andrew Howroyd_, Mar 10 2020

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