Metric Dimensions of Metric Spaces over Vector Groups

Yiming Lei (), Zhongrui Wang and Bing Dai
Additional contact information
Yiming Lei: College of Mathematical Sciences, Bohai University, Jinzhou 121013, China
Zhongrui Wang: College of Mathematical Sciences, Bohai University, Jinzhou 121013, China
Bing Dai: College of Mathematical Sciences, Bohai University, Jinzhou 121013, China

Mathematics, 2025, vol. 13, issue 3, 1-14

Abstract: Let ( X , ρ ) be a metric space. A subset A of X resolves X if every point x ∈ X is uniquely identified by the distances ρ ( x , a ) for all a ∈ A . The metric dimension of ( X , ρ ) is the minimum integer k for which a set A of cardinality k resolves X . We consider the metric spaces of Cayley graphs of vector groups over Z . It was shown that for any generating set S of Z , the metric dimension of the metric space X = X ( Z , S ) is, at most, 2 max S . Thus, X = X ( Z , S ) can be resolved by a finite set. Let n ∈ N with n ≥ 2 . We show that for any finite generating set S of Z n , the metric space X = X ( Z n , S ) cannot be resolved by a finite set.

Keywords: Cayley graph; metric dimension; metric space; vector group (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/3/462/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/3/462/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:3:p:462-:d:1580144

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().