cp's OEIS Frontend

A006821 Number of connected regular graphs of degree 5 (or quintic graphs) with 2n nodes.

%I A006821 M3168 #49 Feb 16 2025 08:32:30
%S A006821 1,0,0,1,3,60,7848,3459383,2585136675,2807105250897,4221456117363365,
%T A006821 8516994770090547979,22470883218081146186209,
%U A006821 75883288444204588922998674,322040154704144697047052726990
%N A006821 Number of connected regular graphs of degree 5 (or quintic graphs) with 2n nodes.
%D A006821 CRC Handbook of Combinatorial Designs, 1996, p. 648.
%D A006821 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 A006821 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A006821 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006821 Leonard Chidiebere Eze, Robert Jajcay, and Jorik Jooken, <a href="https://arxiv.org/abs/2411.19023">On (k,g)-Graphs without (g+1)-Cycles</a>, arXiv:2411.19023 [math.CO], 2024. See p. 18.
%H A006821 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 A006821 Denis S. Krotov, <a href="https://arxiv.org/abs/2012.00038">[[2,10],[6,6]]-equitable partitions of the 12-cube</a>, arXiv:2012.00038 [math.CO], 2020.
%H A006821 Markus Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html">Tables of Regular Graphs</a>
%H A006821 Markus Meringer, <a href="https://www.researchgate.net/publication/228775390_Fast_generation_of_regular_graphs_and_construction_of_cages">Fast generation of regular graphs and construction of cages</a>, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
%H A006821 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuinticGraph.html">Quintic Graph</a>
%H A006821 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RegularGraph.html">Regular Graph</a>
%F A006821 a(n) = A184953(n) + A058275(n).
%F A006821 a(n) = A165626(n) - A165655(n).
%F A006821 Inverse Euler transform of A165626.
%e A006821 a(0)=1 because the null graph (with no vertices) is vacuously 5-regular and connected.
%Y A006821 Contribution (almost all) from _Jason Kimberley_, Feb 10 2011: (Start)
%Y A006821 5-regular simple graphs: this sequence (connected), A165655 (disconnected), A165626 (not necessarily connected).
%Y A006821 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).
%Y A006821 Connected 5-regular simple graphs with girth at least g: this sequence (g=3), A058275 (g=4), A205295 (g=5).
%Y A006821 Connected 5-regular simple graphs with girth exactly g: A184953 (g=3), A184954 (g=4), A184955 (g=5).
%Y A006821 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)
%K A006821 nonn,nice,hard,more
%O A006821 0,5
%A A006821 _N. J. A. Sloane_
%E A006821 By running M. Meringer's GENREG for about 2 processor years at U. Newcastle, a(9) was found by _Jason Kimberley_, Nov 24 2009
%E A006821 a(10)-a(14) from _Andrew Howroyd_, Mar 10 2020