Characterization of geodesic distance on infinite graphs

TL;DR

This paper extends the characterization of geodesic distances from finite to infinite connected graphs and shows that any integer-distance metric space can be embedded into such graph metric spaces.

Contribution

It generalizes the isometry characterization to infinite graphs and demonstrates universal embedding of integer-distance metric spaces.

Findings

Characterization of geodesic distances on infinite graphs

Every integer-distance metric space can be embedded into a graph metric space

Extension of finite graph results to infinite cases

Abstract

Let $G$ be a connected graph and let $d_G$ be the geodesic distance on $V(G)$. The metric spaces $(V(G), d_{G})$ are characterized up to isometry for all finite connected $G$ by David C. Kay and Gary Chartrand in 1964. The main result of the paper expands this characterization on the infinite connected graphs. We also prove that every metric space with integer distances between its points admits an isometric embedding into $(V(G), d_G)$ for suitable $G$.