  For a weighted directed multigraph, let $f_{ij}$ be the total weight of
spanning converging forests that have vertex $i$ in a tree converging to $j$.
We prove that $f_{ij} f_{jk} = f_{ik} f_{jj}$ if and only if every directed
path from $i$ to $k$ contains $j$ (a graph bottleneck equality). Otherwise,
$f_{ij} f_{jk} < f_{ik} f_{jj}$ (a graph bottleneck inequality). In a companion
paper (P. Chebotarev, A new family of graph distances, arXiv preprint
arXiv:0810.2717}. Submitted), this inequality underlies, by ensuring the
triangle inequality, the construction of a new family of graph distances. This
stems from the fact that the graph bottleneck inequality is a multiplicative
counterpart of the triangle inequality for proximities.
