# Find number of edges given number of vertices and average path length?

A while ago I found this equation, not sure where, for determining the number of vertices $$v$$ given an average "edges per node" $$r$$ and an average path length $$n$$. This assumes a DAG where there's one root node and there are no orphans.

$$v = \frac{r^{t-1}\left(1 - r^n\right)}{1 - r}$$

where $$t$$ = starting path level (usually 1), $$n$$ = average path length, and $$r$$ is average outgoing edges per vertex.

I also figured out how to solve for $$n$$ using a logarithmic identity, but is it possible to solve for $$r$$? If I have the average path length and the total number of vertices, can I solve for number of outgoing edges per vertex? I can do it with interpolation and a program, just curious if a derived equation is possible.

• What is an orphan ? Mar 24, 2022 at 8:13
• I was trying to think of a clear way to explain that it's one DAG. The root node has paths to every other node. No other nodes are parentless. Mar 25, 2022 at 22:21