[@KeithEPeterson](https://forum.azimuthproject.org/discussion/comment/18803/#Comment_18803)
I presume you mean...

**Puzzle 104+**. Given an arbitrary connected graph having a node labeled 'x', how many paths of length 'n' go from 'x' to 'x' in that graph?
Where the resulting function \$$H: \mathbb{N} \rightarrow \mathbb{N} \$$ is used to produce a sequence.
For which sequences is it possible to construct a graph which encodes the sequence?

[Edit: Make the question more general](https://forum.azimuthproject.org/discussion/comment/18810/#Comment_18810)

[Edit: Make it a puzzle](https://forum.azimuthproject.org/discussion/comment/18805/#Comment_18805)

[Edit: Strike monotonic](https://forum.azimuthproject.org/discussion/comment/18810/#Comment_18810)

**Edit**
Stated another way...
Given a function as defined above...
$J : \textbf{Graph} \rightarrow ( \mathbb{N} \rightarrow \mathbb{N} )$
Does \$$J\$$ have an inverse? Does \$$J\$$ have a right adjoint?
$J^{-1} : ( \mathbb{N} \rightarrow \mathbb{N} ) \rightarrow \textbf{Graph}$