  The matrix pencil (A,B) = {tB-A | t \in C} is considered under the
assumptions that A is entrywise nonnegative and B-A is a nonsingular M-matrix.
As t varies in [0,1], the Z-matrices tB-A are partitioned into the sets L_s
introduced by Fiedler and Markham. As no combinatorial structure of B is
assumed here, this partition generalizes some of their work where B=I. Based on
the union of the directed graphs of A and B, the combinatorial structure of
nonnegative eigenvectors associated with the largest eigenvalue of (A,B) in
[0,1) is considered.
