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

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 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: 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?

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.