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 $$\theta^2mod\ 2=0$$ i.e. which $\theta$ are nilpotent of order 2 over $\mathbb{Z}_2$?

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 $\mathbb{Z}_2$?