Metrization of weighted graphs

TL;DR

This paper characterizes when a weighted graph's edge weights can be extended to a pseudometric on vertices, identifies the greatest such extension as the shortest-path pseudometric, and describes conditions for the existence of a least extension.

Contribution

It provides necessary and sufficient conditions for weight extension to pseudometrics, characterizes the greatest extension, and identifies when a least extension exists with an explicit formula.

Findings

Shortest-path pseudometric is the greatest extension.

Least extension exists iff the graph is complete k-partite.

Explicit formula for the least extension in special cases.

Abstract

We find a set of necessary and sufficient conditions under which the weight $w:E\to\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric $d:V\times V\to\mathbb R^+$. If these conditions hold and $G$ is a connected graph, then the set $\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the greatest element of $\mathfrak M_w$ with respect to the partial ordering $d_1 \leqslant d_2$ if and only if $d_1(u,v) \leqslant d_2(u,v)$ for all $u,v\in V$. It is shown that every nonvoid poset $(\mathfrak M_w,\leqslant)$ contains the least element $\rho_{0,w}$ if and only if $G$ is a complete $k$-partite graph with $k\geqslant 2$ and in this case the explicit formula for computation of $\rho_{0,w}$ is obtained.