A033483 Number of disconnected 4-valent (or quartic) graphs with n nodes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 8, 25, 88, 378, 2026, 13351, 104595, 930586, 9124662, 96699987, 1095469608, 13175272208, 167460699184, 2241578965849, 31510542635443, 464047929509794, 7143991172244290, 114749135506381940, 1919658575933845129, 33393712487076999918, 603152722419661386031

Offset: 0

Author

Ronald C. Read

Keywords

• nonn
• nice
• hard

References

• R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Links

• Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Disconnected Graph
• Eric Weisstein's World of Mathematics, Quartic Graph

Crossrefs

4-regular simple graphs: A006820 (connected), this sequence (disconnected), A033301 (not necessarily connected). - Jason Kimberley, Jan 08 2011

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), this sequence (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).

Disconnected 4-regular simple graphs with girth at least g: this sequence (g=3), A185244 (g=4), A185245 (g=5), A185246 (g=6).

Programs

Mathematica

A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] - A006820 (* Jean-François Alcover, Jan 21 2020, updated Mar 17 2020 *)

Formula

a(n) = A033301(n) - A006820(n) = Euler_transformation(A006820) - A006820.

a(n) = A068933(n, 4). - Jason Kimberley, Sep 27 2009 and Jan 08 2011

Extensions

Terms a(16)-a(18) from Martin Fuller, Dec 04 2006

Terms a(19)-a(26) from Jason Kimberley, Sep 27 2009 and Dec 30 2010

Terms a(27)-a(33), due to the extension of A006820 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020