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?



Directed graph whose adjacency matrix admits only 0 as eigenvalue
$0$ 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.


