On the metric dimension of barcycentric subdivision of Cayley graphs $Cay(Z_{n}\oplus Z_{m})$
Muhammad Imran, Ali Hasan Ahmad, Omar Saeed Al-Mushayt, Syed Ahtsham Ul Haq Bokhary · Miskolc mathematical notes/Mathematical notes · 2015
D fw 1 ; w 2 ; : : : ; w k g be an ordered set of vertices of G and let v be a vertex of G.The representation r.vjW / of v with respect to W is the k-tuple .d.v; w 1 /; 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 ˚Zm /.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 ˚Zm /.