MMN-1192

Let $W = \{w_1,w_2, \dots ,w_k\}$ be an ordered set of vertices of $G$ and let $v$ be a vertex of $G$. The \emph{representation} $r(v|W)$ of $v$ with respect to $W$ is the k-tuple $(d(v,w_1), d(v,w_2), \dots , d(v,w_k))$. $W$ is called a {\it resolving set} or a {\it 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 \emph{metric basis} for $G$ and this cardinality is the {\it 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 barcycentric subdivision of Cayley graphs $Cay(Z_{n}\oplus Z_{m})$. 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 $Cay(Z_{n}\oplus Z_{m})$.

Vol. 16 (2015), No. 2, pp. 637-646

DOI: 10.18514/MMN.2015.1192

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