A new discrete Hopf-Rinow theorem
- 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 essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating 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.
| Author details: | Matthias KellerORCiDGND, Florentin MünchGND |
|---|---|
| DOI: | https://doi.org/10.1016/j.disc.2019.03.014 |
| ISSN: | 0012-365X |
| ISSN: | 1872-681X |
| Title of parent work (English): | Discrete Mathematics |
| Publisher: | Elsevier |
| Place of publishing: | Amsterdam |
| Publication type: | Article |
| Language: | English |
| Year of first publication: | 2019 |
| Publication year: | 2019 |
| Release date: | 2020/11/18 |
| Volume: | 342 |
| Issue: | 9 |
| Number of pages: | 7 |
| First page: | 2751 |
| Last Page: | 2757 |
| Funding institution: | DFGGerman Research Foundation (DFG) [SPP2026]; Studienstiftung des deutschen Volkes |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
| DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Peer review: | Referiert |
