ELA, Volume 9, pp. 255-269, September 2002, abstract.  

The Combinatorial Structure of Eventually Nonnegative Matrices

Sarah Carnochan Naqvi and Judith J. McDonald

In this paper it is shown that an eventually nonnegative 
matrix A whose index of zero is less than or equal to one, 
exhibits many of the same combinatorial properties as a 
nonnegative matrix. In particular, there is a positive integer
g such that A^g is nonnegative, A and A^g have the same 
irreducible classes, and the transitive closure of the 
reduced graph of A is the same as the transitive closure 
of the reduced graph of A^g. In this instance, many of the 
combinatorial properties of nonnegative matrices carry over
to this subclass of the eventually nonnegative matrices.