On the metric dimension of barycentric subdivision of cayley graphs Cay (Zn⊕Zm)

A. Ahmad, M. Imran, O. Al-Mushayt, S. A.U.H. Bokhary

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

Let W = (w1, w2, .... wk) be an ordered set of vertices of G and let ν be a vertex of G. The representation r(ν|W) of ν with respect to W is the k-tuple (d(ν, w1), d(ν, w2), ..., d(ν, wk)). W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W , or equivalently, if distinct vertices of G have distinct representations with respect to W . A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by dim (G). Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs Cay(Zn⊕Zm). We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of Cayley graphs Cay(Zn⊕Zm).

Original languageEnglish
Pages (from-to)637-646
Number of pages10
JournalMiskolc Mathematical Notes
Volume16
Issue number2
DOIs
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Barycentric subdivision
  • Basis
  • Cayley graph
  • Metric dimension
  • Resolving set

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Numerical Analysis
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

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