The number of connected components in a graph associated with a rectangular (0, 1)-matrix

With a nonzero rectangular (0,1)-matrix A we associate an undirected graph ^GA that corresponds to the linear transformation X→A^XTA. We use the generalized singular-value decomopsition of incidence matrices to count the number of connected components and the number of bipartite connected components of ^GA. We show that ^GA is connected if and only if ^GA has no bipartite connected component or isolated vertex. We also show that if A has no row or column of 0's, then the number of connected components is 12s(s+1) and the number of bipartite connected components is 12s(s-1), where s is the number of chainable components of A, that is, the number of connected components in the undirected graph with adjacency matrix(OA^ATO).