Bipartite Networks have two kinds of nodes, and only connections between unlike nodes.

The equivalent of the adjacency matrix is the incidence matrix, $B$

It can be converted into a unipartite network by a one-mode projection where two vertices are connected if they both have a connection to the same vertex of the other group (we could improve this by adding a weight: the number of those vertices (groups) they have in common).

This projection generally results in an union of cliques, i.e. completely connected components.

The adjacency matrix of the projection is (after we remove the diagonal components) $P=B^TB$ .

One can also have directed bipartite networks (as in Metabolic Networks), and weighted bipartite networks.

Hypergraphs can be represented as bipartite Networks. This is done by mapping the different relations in the hypergraph to a second type of node to which the original nodes can belong by being connected by an edge.