Abstract
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, w1), d(v, w(2)),..., d(v,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 d im(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 C ay(Z(n) circle plus 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 C ay(Z(n) circle plus Z(m)).