[Keith wrote](https://forum.azimuthproject.org/discussion/comment/18803/#Comment_18803):

> Can any strictly increasing monotonic sequence of natural numbers be created from length concatenation of edges of a graph? If not, what is a counterexample?

I have a hunch that the sequence must have a strictly increasing derivative (well, [divided difference](https://en.wikipedia.org/wiki/Finite_difference)) everywhere. For instance, every sequence \\(p_i = 1, 3, 4, \cdots\\) is unobtainable, since in order to go from one null path to three unit-length paths, you must have three self-loops on the start vertex. The next step would multiple those three unit-length paths into nine length-two paths.

**EDIT:** Strictly increasing, not positive. Positive derivative just means monotonic function; I expect that the derivative _itself_ needs to be monotonic, too, as a necessary (but perhaps insufficient) condition.

> Can any strictly increasing monotonic sequence of natural numbers be created from length concatenation of edges of a graph? If not, what is a counterexample?

I have a hunch that the sequence must have a strictly increasing derivative (well, [divided difference](https://en.wikipedia.org/wiki/Finite_difference)) everywhere. For instance, every sequence \\(p_i = 1, 3, 4, \cdots\\) is unobtainable, since in order to go from one null path to three unit-length paths, you must have three self-loops on the start vertex. The next step would multiple those three unit-length paths into nine length-two paths.

**EDIT:** Strictly increasing, not positive. Positive derivative just means monotonic function; I expect that the derivative _itself_ needs to be monotonic, too, as a necessary (but perhaps insufficient) condition.