No abstract

Consider a simple connected undirected graph G = (V G ,E G), where V G represents the vertex set and E G represents the edge set respectively. A subset B of V G is called a resolving set if for every two distinct vertices x, y of G there is a vertex v in set B such that d(x,v) ≠ d(y,v). The resolving set of minimum cardinality is called metric basis of graph G. This minimal cardinality of metric basis is denoted by b(G), and is called metric dimension of G. A subset D of V is called doubly resolving set if for every two vertices x, y of G there are two vertices u, v ∈ D such that d(u,x)-d(u,y) ≠ d(v,x)-d(v,y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by y(G). Some partial cases for metric dimension of circulant graph C n (1,2,3) for n ≥ 12 has been discussed in [21]. Afterwards, problem of finding metric dimension for circulant graph C n (1,2,3), n ≥ 12 has been completely solved by Borchert et al., in [7]. In this paper, we prove that

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