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

1, 0, 0, 1, 3, 60, 7848, 3459383, 2585136675, 2807105250897, 4221456117363365, 8516994770090547979, 22470883218081146186209, 75883288444204588922998674, 322040154704144697047052726990

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Examples

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

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

• Leonard Chidiebere Eze, Robert Jajcay, and Jorik Jooken, On (k,g)-Graphs without (g+1)-Cycles, arXiv:2411.19023 [math.CO], 2024. See p. 18.
• 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.
• Markus Meringer, Tables of Regular Graphs
• Markus 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

Crossrefs

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)

Formula

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

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

Inverse Euler transform of A165626.

Extensions

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