On the metric dimension of circulant graphs

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.