Here's two random $(0,1)$-matrices:
$$
A=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
\end{bmatrix}
\qquad
B=
\begin{bmatrix}
1 & 1 \\
0 & 1 \\
\end{bmatrix}.
$$
They can be interpreted as biadjacency matrices: the graph $G[A]$ has vertices $\{r\_1,r\_2\} \cup \{c\_1,c\_2,c\_3\}$ and undirected edges $r\_i c\_j$ if and only if $A[i,j]=1$, and likewise for $B$.


If we take a [Kronecker product](https://en.wikipedia.org/wiki/Kronecker_product) of $A$ and $B$, we obtain:
$$
A \otimes B=\begin{bmatrix}
\color{red} 1 & \color{red} 1 & 0 & 0 & 0 & 0\\
\color{red} 0 & \color{red} 1 & 0 & 0 & 0 & 0 \\
0 & 0 & \color{blue} 1 & \color{blue} 1 & \color{green} 1 & \color{green} 1 \\
0 & 0 & \color{blue} 0 & \color{blue} 1 & \color{green} 0 & \color{green} 1 \\
\end{bmatrix}
$$
And thus we have a bipartite graph $G[A \otimes B]$ with biadjacency matrix $A \otimes B$.


While I feel like this would have been studied, I don't know what this graph is called. I didn't find it on [Wikipedia's list](https://en.wikipedia.org/wiki/Graph_product): the vertex set is *not* $V(G[A]) \times V(G[B])$.


**Question**: The Kronecker product of two bipartite graphs' biadjacency matrices: what's it called?



The Kronecker product of two bipartite graphs' biadjacency matrices: what's it called?
For bipartite graphs $G[A]$ and $G[B]$, their tensor product $G[A] \times G[B]$ is the disjoint union of bipartite graphs $G[A \otimes B]$ and $G[A \otimes B^T]$. See, for example, p.56 of Hammack, Imrich, Klavžar - Handbook of Product Graphs.


One possible reason why this extensive book doesn't mention this "product" any further is that there's usually no reason to consider $G[A \otimes B]$ over/apart from $G[A \otimes B^T]$ or vice versa in an undirected setting. Were edges of $G[A]$ and $G[B]$ directed from one half to another, their tensor product would consist of $G[A \otimes B]$ (with directed edges) and a bunch of isolated vertices.


