A Graph Bottleneck Inequality

Pavel Chebotarev

paper

A Graph Bottleneck Inequality

arXiv:0810.2732

Abstract

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 $f_{ij} f_{j k} = f_{ik} f_{j j}$ if and only if every directed path from $i$ to $k$ contains $j$ (a graph bottleneck equality). Otherwise, $f_{ij} f_{j k} < f_{ik} f_{j j}$ (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.

6 pages. A revised version