A discrete Hopf-Rinow-theorem

TL;DR

This paper establishes a version of the Hopf-Rinow theorem for discrete spaces using path metrics, introducing the concept of essential local finiteness and characterizing when the resistance metric aligns with the graph structure.

Contribution

It introduces the notion of essential local finiteness for discrete spaces and characterizes the maximal geodesic weight generating the path metric.

Findings

Identifies the necessary condition of essential local finiteness.

Characterizes graphs where the resistance metric is a path metric.

Defines the maximal geodesic weight for complete spaces.

Abstract

We prove a version of the Hopf-Rinow-theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essential local finiteness, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, which generates the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf-Rinow theorem. As an application, we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.