## Abstract

Let W = (w_{1}, w_{2}, .... w_{k}) 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, w_{1}), d(ν, w_{2}), ..., d(ν, w_{k})). 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(Z_{n}⊕Z_{m}). 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(Z_{n}⊕Z_{m}).

Original language | English |
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Pages (from-to) | 637-646 |

Number of pages | 10 |

Journal | Miskolc Mathematical Notes |

Volume | 16 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

## 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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