  We examine the stationary distribution of random walks on directed graphs. In
particular, we focus on the {\em principal ratio}, which is the ratio of
maximum to minimum values of vertices in the stationary distribution. We give
an upper bound for this ratio over all strongly connected graphs on $n$
vertices. We characterize all graphs achieving the upper bound and we give
explicit constructions for these extremal graphs. Additionally, we show that
under certain conditions, the principal ratio is tightly bounded. We also
provide counterexamples to show the principal ratio cannot be tightly bounded
under weaker conditions.
