Two-Regular Graph
A two-regular graph is a regular graph for which all vertex degrees are 2. A two-regular graph consists of one or more (disconnected) cycles.
Two-RegularGraphs
The numbers a_n of two-regular graphs on n=1, 2, ... nodes are 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, ... (OEIS A008483), which are equivalent to the numbers of partitions of n into parts >=3. The first few such graphs are illustrated above.
This sequence has closed form
| a_n=P(n-3)-P(n-2)-P(n-1)+P(n), |
(1)
|
where P(n) is the partition function P. It also has generating function given by
G(x) = [画像:sum_(n=1)^(infty)a_nx^n]
(2)
= [画像:-1+sum_(k=3)^(infty)1/(1-x^k)]
(3)
= [画像:((x-1)^2(x+1))/((x)_infty)-1,]
(4)
where (x)_infty is a q-Pochhammer symbol.
See also
Cycle Graph, Empty Graph, Regular Graph, Vertex DegreeExplore with Wolfram|Alpha
WolframAlpha
References
Sloane, N. J. A. Sequence A008483 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Two-Regular GraphCite this as:
Weisstein, Eric W. "Two-Regular Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Two-RegularGraph.html