## Abstract

Let G=(V,E) be a connected graph and d(x,y) be the distance between the vertices x and yinV(G). A subset of vertices W=^{w1}, ^{w2},⋯,^{wk} is called a resolving set or locating set for G if for every two distinct vertices x,y∈V(G), there is a vertex ^{wi}∈W such that d(x,^{wi})≠d(y,^{wi})fori=1,2, ⋯,k. A resolving set containing the minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension, denoted by dim(G). Let F be a family of connected graphs ^{Gn}:F=(^{Gn})n<_{1} depending on n as follows: the order |V(G)|=φ(n) and lim_{n→∞}φ(n)=∞. If there exists a constant C>0 such that dim(^{Gn})≤C for every n<1 then we shall say that F has bounded metric dimension. The metric dimension of a class of circulant graphs ^{Cn}(1,2) has been determined by Javaid and Rahim (2008) [13]. In this paper, we extend this study to an infinite class of circulant graphs ^{Cn}(1,2,3). We prove that the circulant graphs ^{Cn}(1,2,3) have metric dimension equal to 4 for n≡2,3,4,5(mod6). For n≡0(mod6) only 5 vertices appropriately chosen suffice to resolve all the vertices of ^{Cn}(1,2,3), thus implying that dim( ^{Cn}(1,2,3))≤5 except n≡1(mod6) when dim(^{Cn}(1,2,3)) ≤6.

Original language | English |
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Pages (from-to) | 320-325 |

Number of pages | 6 |

Journal | Applied Mathematics Letters |

Volume | 25 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2012 |

Externally published | Yes |

## Keywords

- Basis
- Circulant graph
- Metric dimension
- Regular graph
- Resolving set

## ASJC Scopus subject areas

- Applied Mathematics