  We associate a signed digraph with a list of matrices whose dimensions permit
them to be multiplied, and whose product is square. Cycles in this graph have a
parity, that is, they are either even (termed e-cycles) or odd (termed
o-cycles). The absence of e-cycles in the graph is shown to imply that the
matrix product is a P0-matrix, i.e., all of its principal minors are
nonnegative. Conversely, the presence of an e-cycle is shown to imply that
there exists a list of matrices associated with the graph whose product fails
to be a P0-matrix. The results generalise a number of previous results relating
P- and P0-matrices to graphs.

Combinatorics