Bipartite networks

I've been thinking lately about bipartite networks: networks with two kinds of node (A and B), such that every link joins an A-node to a B-node (equivalently, networks containing no odd cycles). Here is an example:


It is known that there are 3 distinct connected bipartite networks with 4 nodes, 5 with 5 nodes, 17 with 6 nodes, etc. There are 105,567,921,675,718,772 with 20 nodes, which is more than a few. There are many real-world examples, such as the human diseasome network of genes linked to associated diseases: