Directed graph whose adjacency matrix admits only 0 as eigenvalue

Question

Let $G$ be a directed graph and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed edge from $P_i$ to $P_j$, ($a_{i,j}=0$ otherwise).

Q. Any characterization for directed graphs whose adjacency matrix admits only 0 as the eigenvalue?

Answer 1

$0$ is the only eigenvalue of $A$ if and only if $A$ is nilpotent, which is equivalent to $A^n=0$. But $A^n=0$ expresses that for all $i$ and $j$, there is no path of length $n$ from $P_i$ to $P_j$. This in turn is equivalent to the graph containing no circles.