A rooted graph is a graph in which one node is labeled in a special way so as to distinguish it from other nodes. The special node is called
the root of the graph. The rooted graphs on 
nodes are isomorphic with the symmetric
relations on 
nodes. The counting polynomial for the number of rooted graphs with 
points is
 |
(1)
|
where 
is the symmetric group 
with an additional element 
appended to each element, 
is its pair group,
and 
the corresponding
cycle index (Harary 1994, p. 186). The first
few cycle indices are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Plugging in 
gives the counting polynomials
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
This gives the array of rooted graphs on 
nodes with 
edges as illustrated in the following table (OEIS A070166).
 |  , 1, 2, ... |
| 1 | 1 |
| 2 | 1, 1 |
| 3 | 1, 2, 2, 1 |
| 4 | 1, 2, 4, 6, 4, 2, 1 |
| 5 | 1, 2, 5, 11, 17, 18, 17, 11, 5, 2, 1 |
Plugging in 
into 
then gives the numbers of rooted graphs on 
, 2, ... nodes as 1, 2, 6, 20, 90, 544, ... (OEIS A000666).
