A014378 Number of connected regular graphs of degree 8 with n nodes.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848873, 214755319657939505395, 187549729101764460261498, 186685399408147545744203815, 210977245260028917322933154987

Offset

0,12

Comments

Since the nontrivial 8-regular graph with the least number of vertices is K_9, there are no disconnected 8-regular graphs with less than 18 vertices. Thus for n<18 this sequence is identical to A180260. - Jason Kimberley, Sep 25 2009 and Feb 10 2011

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.

Links

Table of n, a(n) for n=0..22.

Maciej Demianowicz, Progress in the study of the (non)existence of genuinely unextendible product bases, Quantum Info. Proc. 25 (2026), 67.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g.

Markus Meringer, Tables of Regular Graphs.

Eric Weisstein's World of Mathematics, Connected Graph.

Eric Weisstein's World of Mathematics, Octic Graph.

Eric Weisstein's World of Mathematics, Regular Graph.

Formula

a(n) = A184983(n) + A181154(n).

a(n) = A180260(n) - A165878(n). [corrected by Falk Hüffner, May 24 2026]

This sequence is the inverse Euler transformation of A180260.

Example

a(0)=1 because the null graph (with no vertices) is vacuously 8-regular and connected.

Crossrefs

Contribution (almost all) from Jason Kimberley, Feb 10 2011: (Start)

8-regular simple graphs: this sequence (connected), A165878 (disconnected), A180260 (not necessarily connected).

Connected regular simple graphs A005177 (any degree), A068934 (triangular array), specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), this sequence (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).

Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), A181154 (g=4).

Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3). (End)

Sequence in context: A184983 A184980 A184981 * A180260 A058465 A290984

Adjacent sequences: A014375 A014376 A014377 * A014379 A014380 A014381

Keyword

nonn,hard

Author

N. J. A. Sloane

Extensions

Using the symmetry of A051031, a(15) and a(16) were appended by Jason Kimberley, Sep 25 2009

a(17)-a(22) from Andrew Howroyd, Mar 13 2020

Status

approved