Resistance Bounds for First-Passage Percolation and Maximum Flow

Suppose that each edgeeof a network is assigned a random exponential passage time with meanre. Then the expected first-passage time between two vertices is at least the effective resistance between them for the edge resistances re. Similarly, suppose each edge is assigned a random exponential edge capacity with meance. Then the expected maximum flow between two vertices is at least the effective conductance between them for the edge conductances ce. These inequalities are dual to each other for planar graphs and the second is tight up to a factor of 2 for trees; this has implications for a herd of gnus crossing a river delta.