Enumerating programs as graphs, or executing graphs? Both programs and simple graphs can be enumerated using natural numbers (positive integers).

However, what languages or ideas most naturally combine the two?

For example enumerating programs using graphs instead of natural numbers.

Or executing graphs, in other words code described by a graphs shape instead of by text, and not just for tree shaped 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 $S_2$.