On the metric dimension of barycentric subdivision of cayley graphs Cay (Z_n⊕Z_m)

Let W = (w₁, w₂, .... 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, w₁), d(ν, w₂), ..., 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).