My task is to prove that for a connected bipartite graph, each of its adjacency matrix powers $A,A^2,A^3,\dots$ contains some zeros.
I started by doing the following:
The adjacency matrix has the form $$A=\begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix},$$ where $B$ contains ones and zeros, and only contains ones for $K_{m,n}$, a complete bipartite graph. Clearly $A$ contains zeros. Now,
$$A^2=\begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix}\begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix}=\begin{pmatrix} BB^T & 0 \\ 0 & B^TB \end{pmatrix}$$ and
$$A^3=\begin{pmatrix} BB^T & 0 \\ 0 & B^TB \end{pmatrix}\begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix}=\begin{pmatrix} 0 & BB^TB \\ B^TB & 0 \end{pmatrix}.$$ It seems like this patterns continues, with the zeros swapping between the diagonal and off-diagonal. So for $k$ even, the zeros are on the off-diagonal of $A^k$ and for $k$ odd, the zeros are on the diagonal. I'm not really sure this is enough to justify a concrete proof, so some guidance would be helpful!