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 = {w(1), w(2), ..., w(k)} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v vertical bar W) of v with respect to W is the k-tuple (d(v, w(1)), d(v, w(2)), ..., 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 13(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 e 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 e Z2).