  We consider certain matrix-products where successive matrices in the product
belong alternately to a particular qualitative class or its transpose. The main
theorems relate structural and spectral properties of these matrix-products to
the structure of underlying bipartite graphs. One consequence is a
characterisation of caterpillars: a graph is a caterpillar if and only if all
matrix-products associated with it have real nonnegative spectrum. Several
other equivalences of this kind are proved. The work is inspired by certain
questions in dynamical systems where such products arise naturally as Jacobian
matrices, and the results have implications for the existence and stability of
equilibria in these systems.
