MMN-1192
On the metric dimension of barcycentric subdivision of Cayley graphs $Cay(Z_{n}\oplus Z_{m})$
Ali Ahmad; Muhammad Imran; O. Al- Mushayt; Syed Ahtsham Ul Haq Bokhary;Abstract
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