On the metric dimension of circulant graphs

Muhammad Imran, A. Q. Baig, Syed Ahtsham Ul Haq Bokhary, Imran Javaid

Research output: Contribution to journalArticlepeer-review

51 Citations (Scopus)

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 limn→∞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 languageEnglish
Pages (from-to)320-325
Number of pages6
JournalApplied Mathematics Letters
Volume25
Issue number3
DOIs
Publication statusPublished - Mar 2012
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics

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