ELA, Volume 7, pp. 30-40, March 2000, abstract.

Digraphs with large exponent

S. Kirkland, D. D. Olesky, and P. van den Driessche

Primitive digraphs on n vertices with exponents at least
floor(w_n / 2) + 2 are considered, where w_n = (n-1)(n-1) +1, 
and where floor(x) is the greatest integer less than or equal to x.  
For n greater than or equal to 3, all such digraphs containing a 
Hamilton cycle are characterized; and for n greater than or equal to 6, 
all such digraphs containing a cycle of length n-1 are characterized.
Each eigenvalue of any stochastic matrix having a digraph
in one of these two classes is proved to be geometrically simple.