On the metric dimension of barycentric subdivision of Cayley graphs

In a connected graph G, the distance d(u, v) denotes the distance between two vertices u and v of G. Let W = {w₁, w₂, ···, w_k} be an ordered set of vertices of G and let v be a vertex of G. The representationr(v|W) of v with respect to W is the k-tuple (d(v, w₁), d(v,w₂), ···, d(v, w_k)). The set 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 β(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 ⨁ Z₂). 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 barycentric subdivision of Cayley graphs Cay (Z_n ⨁ Z₂).