Characterization of nilpotent adjacency matrices Let $\theta$ be the adjacency matrix of a simple graph (symmetric and zeros on the diagonal). What is the characterization of those $\theta$ which satisfy $$\theta^2 \equiv 0 \pmod{2}$$ i.e. which $\theta$ are nilpotent of order 2 over $\mathbb{Z}_2$?
 A: I do not think you will find a characterisation that is substantially better than the one suggested by LSpice, because this class includes other classes of graphs for which characterisations are unknown.
In particular, a $(0,2)$-graph is a graph such that each pair of distinct vertices has exactly 0 or 2 common neighbours. So if $A$ is the adjacency matrix of a $(0,2)$-graph, then every off-diagonal entry of $A^2$ is either $0$ or $2$ and hence even. 
The diagonal entries of $A^2$ are the vertex degrees, and as $(0,2)$-graphs are regular, the diagonal is constant. So your class includes "even degree $(0,2)$-graphs" which are not known.
Of course there are other graphs that are not $(0,2)$-graphs that still have the nilpotent property. For example, take the prism over the complete graph of even order, i.e., the Cartesian product $K_2 \square K_r$. Then every pair of vertices in the same copy of $K_r$ are connected by $r-2$ paths of length $2$, which is even. Now consider vertices in different copies of $K_r$ — if the vertices are adjacent then there are no paths of length $2$ between them, while if the vertices are not adjacent, then there are exactly $2$ paths of length between them. In addition, these graphs do not have any twin vertices.
There may be a few more things that can be said about such graphs, but almost certainly not a complete characterisation.
