Let the adjacency matrix of an *undirected bipartite* graph be $A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix}$ where B is called the *biadjacency matrix*.

Now, by instead interpreting B as an *adjacency matrix* of a *not necessarily bipartite (arbitrary) directed* graph, we get a bijection between the bipartite undirected graph (with *bi*adjacency matrix B) and the arbitrary directed graph (with adjacency matrix B).

Does this bijection have a name? Is it discussed somewhere in literature? Thank you!

meanis simply 'we get a map $\textsf{UndirectedBipartiteGraphs}$ $\to$ $\textsf{DirectedGraphs}$. I don't see how 'equivalence relation' makes any sense here. On what set should this relation be defined? How should it even bereflexive? Would you please clarify and possiblyedit the question accordingly? Also, saying "non-bipartite" is wrong: this directed graph canhappento be bipartite. The correct way to put is "anot necessarily bipartite directedgraph". $\endgroup$ – Peter Heinig Feb 26 '18 at 8:52