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We examine the stationary distribution of random walks on directed graphs. Inparticular, we focus on the {\em principal ratio}, which is the ratio ofmaximum to minimum values of vertices in the stationary distribution. We givean upper bound for this ratio over all strongly connected graphs on $n$vertices. We characterize all graphs achieving the upper bound and we giveexplicit constructions for these extremal graphs. Additionally, we show thatunder certain conditions, the principal ratio is tightly bounded. We alsoprovide counterexamples to show the principal ratio cannot be tightly boundedunder weaker conditions.

Extreme values of the stationary distribution of random walks on directed graphs