<s>[INST] Generating function for non-isomorphic regular graphs. Determine a generating function for the number of non-isomorphic (n−2)-regular graphs of order n, for n ≥ 2. 
I've been staring at this for hours and can't find a place to start, any help would be appreciated.
 [/INST] We enumerate the complements, namely non-isomorphic 2-regular graphs.
These are sets of cycles and we find
$$\prod_{k\ge 3} \frac{1}{1-z^k}
= (1-z)(1-z^2) \prod_{k\ge 1} \frac{1}{1-z^k}.$$
The  term   $1/(1-z^k)$  is   the  OGF  of   zero,  one,   two,  three
etc. instances of a cycle of order $k.$ 
This is the OGF
$$(1-z-z^2+z^3) \prod_{k\ge 1} \frac{1}{1-z^k}.$$
Using the partition function we get for $n\ge 3$
$$p(n) - p(n-1) - p(n-2) + p(n-3).$$
We obtain the sequence
$$1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13,
\\ 17, 21, 25, 33, 39, 49, \ldots$$
which  points  us  to OEIS  A00843,  where
these  data are  confirmed  (indeed we  have non-isomorphic  2-regular
graphs). 
For  the case  where the  graphs  are labeled  we again  have sets  of
cycles (with dihedral symmetry):
$$\def\textsc#1{\dosc#1\csod}
\def\dosc#1#2\csod{{\rm #1{\small #2}}}
\textsc{SET}(\textsc{DHD}_{=3}(\mathcal{Z})
+ \textsc{DHD}_{=4}(\mathcal{Z})
+ \textsc{DHD}_{=5}(\mathcal{Z})
+ \cdots).$$
This gives the EGF
$$\exp
\left(\frac{1}{2} \frac{z^3}{3}
+\frac{1}{2} \frac{z^4}{4}
+\frac{1}{2} \frac{z^5}{5}+\cdots\right)
\\ = \exp\left(-\frac{1}{2} z - \frac{1}{2} \frac{z^2}{2}
+ \frac{1}{2} \log\frac{1}{1-z}\right)
\\ = \frac{1}{\sqrt{1-z}}
\exp\left(-\frac{z}{2}-\frac{z^2}{4}\right).$$
We get the sequence starting at $n\ge 3$:
$$1, 3, 12, 70, 465, 3507, 30016, 286884, 3026655,
\\ 34944085, 438263364, 5933502822, 86248951243, \ldots$$
which points us to  OEIS A001205, where we
get confirmation of these data once more.
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