A Graph Bottleneck Inequality

TL;DR

This paper establishes a graph bottleneck inequality involving spanning forests in directed multigraphs, which underpins a new family of graph distances satisfying the triangle inequality.

Contribution

It proves a novel multiplicative inequality characterizing graph bottlenecks, foundational for constructing new graph distances with triangle inequality properties.

Findings

Proves the graph bottleneck inequality and equality conditions.

Links the inequality to the construction of graph distances.

Shows the inequality as a multiplicative analogue of the triangle inequality.

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 $f_{ij} f_{jk} = f_{ik} f_{jj}$ 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.