bipartite network graph can have block diagonal matrix?

Question

This is a bipartite network. Image

If you create it's adjacency matrix you see that it's diagonal contains zero elements, and two blocks.

is it a block-diagonal matrix? I can see two blocks, but I think that the graph is one connected component and not two (two blocks).

Answer 1

For block-diagonal form, the blocks must be along the main diagonal. In general, the adjacency matrix of a graph is block-diagonal with one block per connected component. For your example, the matrix is block-diagonal with one block but not with two blocks.