Enumerating programs as graphs, or executing graphs?
We can summarize your original findings as follows: there is an effective enumeration for programs and graphs. An effective enumeration of a countable set $S$ is a computable bijection between the natural numbers and $S$, that is, a program $P(n)$ such that $S = \{ P(n) : n \geq 0 \}$ and $P(n) \neq P(m)$ for $n \neq m$.

There are many countable sets which can be effectively enumerated – there is nothing special about programs and graphs. We can generalize and say that if $S_1,S_2$ can be effectively enumerated, then there is a computable bijection between $S_1$ and $S_2$ (why?), but this doesn't say much about any relation between $S_1$ and $S_2$.