A two-regular graph is a regular graph for which all local degrees are 2. A two-regular graph consists of one or more (disconnected) cycles.
The numbers 
of two-regular graphs on 
, 2, ... nodes are 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, ... (OEIS
A008483), which are equivalent to the numbers
of partitions of 
into parts 
.
The first few such graphs are illustrated above.
This sequence has closed form
 |
(1)
|
where 
is the partition function P. It also has generating function given by
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a q-Pochhammer symbol.
