Let $G$ be a digraph such that there is an unique directed path of length $k$ between any two vertices.
Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $1$.

Then it is not too hard to show, using algebraic methods, that the number of vertices is $d^k$ for some integer $d$, that each vertex has indegree and outdegree $d$ and that $G$ has $d$ loops. Let's call such a digraph a $(d,k)$*-nice* digraph.

A simple example of a $(d,k)$-nice digraph is the de Bruijn graph for words of lenght $k$ over an alphabet of $d$ symbols. Note also that, if $G$ is a $(d,k)$-nice digraph, then the line digraph of $G$ is a $(d,k+1)$-nice digraph.

There are however, other examples than de Bruijn graphs. The following digraph, for example, is $(3,2)$-nice: http://graphonline.ru/en/?graph=iuDxicdebMgXCAFE. This example is, unfortunately, very asymmetric and doesn't seem to have a simple interpretation like de Bruijn graphs.

My questions are:
- Does this class of digraphs have already been studied? 
- Is there a way to classify all $(d,k)$-nice graphs?
- If there is no simple classification in the general case (which seems plausible given the irregular example I gave), can we hope to have a classification for specific values of $d$? In particular, can we find examples of $(2,k)$-nice digraphs that are not de Bruijn?
- Are there any other interesting properties that we can prove that these digraphs have?