I'm trying to determine if there are properties of certain incidence matrices that mean that their multiple will create a valid incidence matrix, i.e., the total number of non zero entries in each row is either 2 or 0.
In the set of matrices $\mathbb{N}_{2}^{3\times3}$, there are three valid columns: $$\begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}1\\0\\1\end{bmatrix}, \begin{bmatrix}0\\1\\1\end{bmatrix}$$ Which can make $3^{3}$ possible matrices (or $4^{4}$ if the zero vector / 'missing edges' is included) but only some will multiply together to make a third valid incidence matrix.
So in all, I'd like to know if there are any ways to determine if the product of two matrices will preserve the property of being a valid incidence matrix.
It would be highly dependent on how the edges and vertices are ordered. I am struggling a bit to see why this would be something interesting (but that doesn't mean you don't have a reason to be interested in it).
You just need to phrase your question in terms of what the rows/columns of the incidence matrix means, and what it means to multiply matrices... this will give a very convoluted combinatorial condition like "for each vertex $v_{i}$ of the first graph, let $I_{i}$ be the set of edge indices for the edges adjacent to $v_{i}$; then for each edge $e_{j}$ of the second graph, there can be at most one vertex (of the second graph) incident with $e_{j}$ whose index is in $I_{i}$; and furthermore, there must be either $0$ or $2$ edges $e_{j}$ incident with a vertex whose index is in $I_{i}$."